bernoulli number bernoulli numbers bn n fraction decimal 0 1 1000000000 1 12 0500000000 2 16 0166666666 3 0 0000000000 4 130 0033333333 5 0 0000000000 6 142 0023809523 7 0 0000000000 8 130 0033333333 9 0 0000000000 10 566 0075757575 11 0 0000000000 12 6912730 0253113553 13 0 0000000000 14 76 1166666666 15 0 0000000000 16 3617510 7092156862 17 0 0000000000 18 43867798 5497117794 19 0 0000000000 20 174611330 5291242424 graphs of modern bernoulli numbers b circles compared with bernoulli numbers in older literature b crosses in mathematics the bernoulli numbers bn are a sequence of rational numbers which occur frequently in number theory the values of the first 20 bernoulli numbers are given in the adjacent table for every even n other than 0 bn is negative if n is divisible by 4 and positive otherwise for every odd n other than 1 bn 0 the superscript used in this article designates the two sign conventions for bernoulli numbers only the n 1 term is affected bn with b1 12 oeisa027641 oeisa027642 is the sign convention prescribed by nist and most modern textbooks bn with b1 12 oeisa164555 oeisa027642 is sometimes used in the older literature in the formulas below one can switch from one sign convention to the other with the relation b n 1 n b n displaystyle bn1nbn the bernoulli numbers are special values of the bernoulli polynomials b n x displaystyle bnx with b n b n 0 displaystyle bnbn0 and b n b n 1 displaystyle bnbn1 since bn 0 for all odd n 1 and many formulas only involve evenindex bernoulli numbers some authors write bn to mean b2n this article does not follow this notation the bernoulli numbers appear in the taylor series expansions of the tangent and hyperbolic tangent functions in faulhabers formula for the sum of powers of the first positive integers in the eulermaclaurin formula and in expressions for certain values of the riemann zeta function the bernoulli numbers were discovered around the same time by the swiss mathematician jacob bernoulli after whom they are named and independently by japanese mathematician seki kwa sekis discovery was posthumously published in 1712 as a result the bernoulli numbers have the distinction of being the subject of the first published complex computer program history early history the bernoulli numbers are rooted in the early history of the computation of sums of integer powers which have been of interest to mathematicians since antiquity a page from seki takakazus katsuyo sampo 1712 tabulating binomial coefficients and bernoulli numbers methods to calculate the sum of the first n positive integers the sum of the squares and of the cubes of the first n positive integers were known but there were no real formulas only descriptions given entirely in words among the great mathematicians of antiquity to consider this problem were pythagoras c 572497bce greece archimedes 287212bce italy aryabhata b 476 india abu bakr alkaraji d 1019 persia and abu ali alhasan ibn alhasan ibn alhaytham 9651039 iraq during the late sixteenth and early seventeenth centuries mathematicians made significant progress in the west thomas harriot 15601621 of england johann faulhaber 15801635 of germany pierre de fermat 16011665 and fellow french mathematician blaise pascal 16231662 all played important roles thomas harriot seems to have been the first to derive and write formulas for sums of powers using symbolic notation but even he calculated only up to the sum of the fourth powers johann faulhaber gave formulas for sums of powers up to the 17th power in his 1631 academia algebrae far higher than anyone before him but he did not give a general formula blaise pascal in 1654 proved pascals identity relating the sums of the pth powers of the first n positive integers for p 0 1 2 k the swiss mathematician jakob bernoulli 16541705 was the first to realize the existence of a single sequence of constants b0 b1 b2 which provide a uniform formula for all sums of powers knuth 1993 the joy bernoulli experienced when he hit upon the pattern needed to compute quickly and easily the coefficients of his formula for the sum of the cth powers for any positive integer c can be seen from his comment he wrote with the help of this table it took me less than half of a quarter of an hour to find that the tenth powers of the first 1000 numbers being added together will yield the sum 91409924241424243424241924242500 bernoullis result was published posthumously in ars conjectandi in 1713 seki kwa independently discovered the bernoulli numbers and his result was published a year earlier also posthumously in 1712 however seki did not present his method as a formula based on a sequence of constants bernoullis formula for sums of powers is the most useful and generalizable formulation to date the coefficients in bernoullis formula are now called bernoulli numbers following a suggestion of abraham de moivre bernoullis formula is sometimes called faulhabers formula after johann faulhaber who found remarkable ways to calculate sum of powers but never stated bernoullis formula to call bernoullis formula faulhabers formula does injustice to bernoulli and simultaneously hides the genius of faulhaber as faulhabers formula is in fact more efficient than bernoullis formula according to knuth knuth 1993 a rigorous proof of faulhabers formula was first published by carl jacobi in 1834 jacobi 1834 knuths indepth study of faulhabers formula concludes the nonstandard notation on the lhs is explained further on faulhaber never discovered the bernoulli numbers ie he never realized that a single sequence of constants b0 b1 b2 would provide a uniform n m 1 m 1 b 0 n m 1 m 1 1 b 1 n m m 1 2 b 2 n m 1 m 1 m b m n displaystyle quad sum nmfrac 1m1leftb0nm1binom m11b1nmbinom m12b2nm1cdots binom m1mbmnright or n m 1 m 1 b 0 n m 1 m 1 1 b 1 n m m 1 2 b 2 n m 1 1 m m 1 m b m n displaystyle quad sum nmfrac 1m1leftb0nm1binom m11b1nmbinom m12b2nm1cdots 1mbinom m1mbmnright for all sums of powers he never mentioned for example the fact that almost half of the coefficients turned out to be zero after he had converted his formulas for nm from polynomials in n to polynomials in n knuth 1993 p14 reconstruction of summae potestatum jakob bernoullis summae potestatum 1713 the bernoulli numbers oeisa164555noeisa027642n were introduced by jakob bernoulli in the book ars conjectandi published posthumously in 1713 page 97 the main formula can be seen in the second half of the corresponding facsimile the constant coefficients denoted a b c and d by bernoulli are mapped to the notation which is now prevalent as a b2 b b4 c b6 d b8 the expression cc1c2c3 means cc1c2c3 the small dots are used as grouping symbols using todays terminology these expressions are falling factorial powers ck the factorial notation k as a shortcut for 1 2 k was not introduced until 100 years later the integral symbol on the left hand side goes back to gottfried wilhelm leibniz in 1675 who used it as a long letter s for summa sum the mathematics genealogy project shows leibniz as the academic advisor of jakob bernoulli see also the earliest uses of symbols of calculus the letter n on the left hand side is not an index of summation but gives the upper limit of the range of summation which is to be understood as 1 2 n putting things together for positive c today a mathematician is likely to write bernoullis formula as k 1 n k c n c 1 c 1 1 2 n c k 2 c b k k c k 1 n c k 1 displaystyle sum k1nkcfrac nc1c1frac 12ncsum k2cfrac bkkcunderline k1nck1 this formula suggests setting b1 12 when switching from the socalled archaic enumeration which uses only the even indices 2 4 6 to the modern form more on different conventions in the next paragraph most striking in this context is the fact that the falling factorial ck1 has for k 0 the value 1c 1 thus bernoullis formula can be written k 1 n k c k 0 c b k k c k 1 n c k 1 displaystyle sum k1nkcsum k0cfrac bkkcunderline k1nck1 if b1 12 recapturing the value bernoulli gave to the coefficient at that position note that the formula for k 1 n k 9 displaystyle textstyle sum k1nk9 in the first half contains an error at the last term it must be 3 20 n 2 displaystyle tfrac 320n2 instead of 1 12 n 2 displaystyle tfrac 112n2 definitions many characterizations of the bernoulli numbers have been found in the last 300 years and each could be used to introduce these numbers here only three of the most useful ones are mentioned a recursive equation an explicit formula a generating function for the proof of the equivalence of the three approaches see ireland rosen 1990 or conway guy 1996 recursive definition the bernoulli numbers obey the sum formulas k 0 m m 1 k b k m 0 k 0 m m 1 k b k m 1 displaystyle beginalignedsum k0mbinom m1kbkdelta m0sum k0mbinom m1kbkm1endaligned where m 0 1 2 displaystyle m012 and denotes the kronecker delta solving for b m displaystyle bmmp gives the recursive formulas b m m 0 k 0 m 1 m k b k m k 1 b m 1 k 0 m 1 m k b k m k 1 displaystyle beginalignedbmdelta m0sum k0m1binom mkfrac bkmk1bm1sum k0m1binom mkfrac bkmk1endaligned explicit definition in 1893 louis saalschtz listed a total of 38 explicit formulas for the bernoulli numbers saalschtz 1893 usually giving some reference in the older literature one of them is b m k 0 m v 0 k 1 v k v v m k 1 b m k 0 m v 0 k 1 v k v v 1 m k 1 displaystyle beginalignedbmsum k0msum v0k1vbinom kvfrac vmk1bmsum k0msum v0k1vbinom kvfrac v1mk1endaligned generating function the exponential generating functions are t e t 1 t 2 coth t 2 1 m 0 b m t m m t 1 e t t 2 coth t 2 1 m 0 b m t m m displaystyle beginalignedfrac tet1frac t2leftoperatorname coth frac t21rightsum m0infty frac bmtmmfrac t1etfrac t2leftoperatorname coth frac t21rightsum m0infty frac bmtmmendaligned the normal generating function z 1 1 z 1 m 0 b m z m displaystyle z1psi 1z1sum m0infty bmzm is an asymptotic series it contains the trigamma function 1 bernoulli numbers and the riemann zeta function the bernoulli numbers as given by the riemann zeta function the bernoulli numbers can be expressed in terms of the riemann zeta function bn n1 n for n 1 here the argument of the zeta function is 0 or negative by means of the zeta functional equation and the gamma reflection formula the following relation can be obtained b 2 n 1 n 1 2 2 n 2 2 n 2 n displaystyle b2nfrac 1n122n2pi 2nzeta 2n for n 1 now the argument of the zeta function is positive it then follows from 1 n and stirlings formula that b 2 n 4 n n e 2 n displaystyle b2nsim 4sqrt pi nleftfrac npi eright2n for n efficient computation of bernoulli numbers in some applications it is useful to be able to compute the bernoulli numbers b0 through bp 3 modulo p where p is a prime for example to test whether vandivers conjecture holds for p or even just to determine whether p is an irregular prime it is not feasible to carry out such a computation using the above recursive formulae since at least a constant multiple of p2 arithmetic operations would be required fortunately faster methods have been developed buhler et al 2001 which require only op log p2 operations see big o notation david harvey harvey 2008 describes an algorithm for computing bernoulli numbers by computing bn modulo p for many small primes p and then reconstructing bn via the chinese remainder theorem harvey writes that the asymptotic time complexity of this algorithm is on2 logn2 and claims that this implementation is significantly faster than implementations based on other methods using this implementation harvey computed bn for n 108 harveys implementation has been included in sagemath since version 31 prior to that bernd kellner kellner 2002 computed bn to full precision for n 106 in december 2002 and oleksandr pavlyk pavlyk 2008 for n 107 with mathematica in april 2008 computer year n digits j bernoulli 1689 10 1 l euler 1748 30 8 j c adams 1878 62 36 d e knuth t j buckholtz 1967 70031672000000000001672 70033330000000000003330 g fee s plouffe 1996 700410000000000000010000 700427677000000000027677 g fee s plouffe 1996 7005100000000000000100000 7005376755000000000376755 b c kellner 2002 70061000000000000001000000 70064767529000000004767529 o pavlyk 2008 700710000000000000010000000 700757675260000000057675260 d harvey 2008 7008100000000000000100000000 7008676752569000000676752569 digits is to be understood as the exponent of 10 when bn is written as a real number in normalized scientific notation applications of the bernoulli numbers asymptotic analysis arguably the most important application of the bernoulli number in mathematics is their use in the eulermaclaurin formula assuming that f is a sufficiently often differentiable function the eulermaclaurin formula can be written as k a b 1 f k a b f x d x k 1 m b k k f k 1 b f k 1 a r f m displaystyle sum kab1fkint abfxdxsum k1mfrac bkkfk1bfk1arfm this formulation assumes the convention b1 12 using the convention b1 12 the formula becomes k a 1 b f k a b f x d x k 1 m b k k f k 1 b f k 1 a r f m displaystyle sum ka1bfkint abfxdxsum k1mfrac bkkfk1bfk1arfm here f0 f ie the zerothorder derivative of a f is just f moreover let f1 denote an antiderivative of f by the fundamental theorem of calculus a b f x d x f 1 b f 1 a displaystyle int abfxdxf1bf1a thus the last formula can be further simplified to the following succinct form of the eulermaclaurin formula k a b f k k 0 m b k k f k 1 b f k 1 a r f m displaystyle sum kabfksum k0mfrac bkkfk1bfk1arfm this form is for example the source for the important eulermaclaurin expansion of the zeta function s k 0 m b k k s k 1 r s m b 0 0 s 1 b 1 1 s 0 b 2 2 s 1 r s m 1 s 1 1 2 1 12 s r s m displaystyle beginalignedzeta ssum k0mfrac bkksoverline k1rsmfrac b00soverline 1frac b11soverline 0frac b22soverline 1cdots rsmfrac 1s1frac 12frac 112scdots rsmendaligned here sk denotes the rising factorial power bernoulli numbers are also frequently used in other kinds of asymptotic expansions the following example is the classical poincartype asymptotic expansion of the digamma function z ln z k 1 b k k z k displaystyle psi zsim ln zsum k1infty frac bkkzk sum of powers faulhabers formula bernoulli numbers feature prominently in the closed form expression of the sum of the mth powers of the first n positive integers for m n 0 define s m n k 1 n k m 1 m 2 m n m displaystyle smnsum k1nkm1m2mcdots nm this expression can always be rewritten as a polynomial in n of degree m 1 the coefficients of these polynomials are related to the bernoulli numbers by bernoullis formula s m n 1 m 1 k 0 m m 1 k b k n m 1 k displaystyle smnfrac 1m1sum k0mbinom m1kbknm1k where m 1k denotes the binomial coefficient for example taking m to be 1 gives the triangular numbers 0 1 3 6 oeisa000217 1 2 n 1 2 b 0 n 2 2 b 1 n 1 1 2 n 2 n displaystyle 12cdots nfrac 12b0n22b1n1tfrac 12n2n taking m to be 2 gives the square pyramidal numbers 0 1 5 14 oeisa000330 1 2 2 2 n 2 1 3 b 0 n 3 3 b 1 n 2 3 b 2 n 1 1 3 n 3 3 2 n 2 1 2 n displaystyle 1222cdots n2frac 13b0n33b1n23b2n1tfrac 13leftn3tfrac 32n2tfrac 12nright some authors use the alternate convention for bernoulli numbers and state bernoullis formula in this way s m n 1 m 1 k 0 m 1 k m 1 k b k n m 1 k displaystyle smnfrac 1m1sum k0m1kbinom m1kbknm1k bernoullis formula is sometimes called faulhabers formula after johann faulhaber who also found remarkable ways to calculate sums of powers faulhabers formula was generalized by v guo and j zeng to a qanalog guo zeng 2005 taylor series the bernoulli numbers appear in the taylor series expansion of many trigonometric functions and hyperbolic functions tangent tan x n 1 1 n 1 2 2 n 2 2 n 1 b 2 n 2 n x 2 n 1 x 2 displaystyle beginalignedtan xsum n1infty frac 1n122n22n1b2n2nx2n1leftxrightfrac pi 2endaligned cotangent cot x 1 x n 0 1 n b 2 n 2 x 2 n 2 n 0 x displaystyle beginalignedcot xfrac 1xsum n0infty frac 1nb2n2x2n2nqquad 0xpi endaligned hyperbolic tangent tanh x n 1 2 2 n 2 2 n 1 b 2 n 2 n x 2 n 1 x 2 displaystyle beginalignedtanh xsum n1infty frac 22n22n1b2n2nx2n1xfrac pi 2endaligned hyperbolic cotangent coth x 1 x n 0 b 2 n 2 x 2 n 2 n 0 x displaystyle beginalignedcoth xfrac 1xsum n0infty frac b2n2x2n2nqquad qquad 0xpi endaligned laurent series the bernoulli numbers appear in the following laurent series digamma function z ln z k 1 b k k z k displaystyle psi zln zsum k1infty frac bkkzk use in topology the kervairemilnors formula for the order of the cyclic group of diffeomorphism classes of exotic 4n 1spheres which bound parallelizable manifolds involves bernoulli numbers let esn be the number of such exotic spheres for n 2 then es n 2 2 n 2 2 4 n 3 numerator b 4 n 4 n displaystyle textit esn22n224n3operatorname numerator leftfrac b4n4nright the hirzebruch signature theorem for the l genus of a smooth oriented closed manifold of dimension 4n also involves bernoulli numbers connections with combinatorial numbers the connection of the bernoulli number to various kinds of combinatorial numbers is based on the classical theory of finite differences and on the combinatorial interpretation of the bernoulli numbers as an instance of a fundamental combinatorial principle the inclusionexclusion principle connection with worpitzky numbers the definition to proceed with was developed by julius worpitzky in 1883 besides elementary arithmetic only the factorial function n and the power function km is employed the signless worpitzky numbers are defined as w n k v 0 k 1 v k v 1 n k v k v displaystyle wnksum v0k1vkv1nfrac kvkv they can also be expressed through the stirling numbers of the second kind w n k k n 1 k 1 displaystyle wnkkleftn1 atop k1right a bernoulli number is then introduced as an inclusionexclusion sum of worpitzky numbers weighted by the harmonic sequence 11213 b n k 0 n 1 k w n k k 1 k 0 n 1 k 1 v 0 k 1 v v 1 n k v displaystyle bnsum k0n1kfrac wnkk1 sum k0nfrac 1k1sum v0k1vv1nk choose v b0 1 b1 1 12 b2 1 32 23 b3 1 72 123 64 b4 1 152 503 604 245 b5 1 312 1803 3904 3605 1206 b6 1 632 6023 21004 33605 25206 7207 this representation has b1 12 consider the sequence sn n 0 from worpitzkys numbers oeisa028246 oeisa163626 applied to s0 s0 s1 s0 s1 s2 s0 s1 s2 s3 is identical to the akiyamatanigawa transform applied to sn see connection with stirling numbers of the first kind this can be seen via the table identity of worpitzkys representation and akiyamatanigawa transform 1 0 1 0 0 1 0 0 0 1 0 0 0 0 1 1 1 0 2 2 0 0 3 3 0 0 0 4 4 1 3 2 0 4 10 6 0 0 9 21 12 1 7 12 6 0 8 38 54 24 1 15 50 60 24 the first row represents s0 s1 s2 s3 s4 hence for the second fractional euler numbers oeisa198631 n oeisa006519 n 1 e0 1 e1 1 12 e2 1 32 24 e3 1 72 124 68 e4 1 152 504 608 2416 e5 1 312 1804 3908 36016 12032 e6 1 632 6024 21008 336016 252032 72064 a second formula representing the bernoulli numbers by the worpitzky numbers is for n 1 b n n 2 n 1 2 k 0 n 1 2 k w n 1 k displaystyle bnfrac n2n12sum k0n12kwn1k the simplified second worpitzkys representation of the second bernoulli numbers is oeisa164555 n 1 oeisa027642n 1 n 12n 2 2 oeisa198631n oeisa006519n 1 which links the second bernoulli numbers to the second fractional euler numbers the beginning is 12 16 0 130 0 142 12 13 314 215 562 121 1 12 0 14 0 12 the numerators of the first parentheses are oeisa111701 see connection with stirling numbers of the first kind connection with stirling numbers of the second kind if skm denotes stirling numbers of the second kind then one has j k m 0 k j m s k m displaystyle jksum m0kjunderline mskm where jm denotes the falling factorial if one defines the bernoulli polynomials bkj as b k j k m 0 k 1 j m 1 s k 1 m m b k displaystyle bkjksum m0k1binom jm1sk1mmbk where bk for k 0 1 2 are the bernoulli numbers then after the following property of binomial coefficient j m j 1 m 1 j m 1 displaystyle binom jmbinom j1m1binom jm1 one has j k b k 1 j 1 b k 1 j k 1 displaystyle jkfrac bk1j1bk1jk1 one also has following for bernoulli polynomials b k j n 0 k k n b n j k n displaystyle bkjsum n0kbinom knbnjkn the coefficient of j in jm 1 is 1mm 1 comparing the coefficient of j in the two expressions of bernoulli polynomials one has b k m 0 k 1 m m m 1 s k m displaystyle bksum m0k1mfrac mm1skm resulting in b1 12 which is an explicit formula for bernoulli numbers and can be used to prove vonstaudt clausen theorem connection with stirling numbers of the first kind the two main formulas relating the unsigned stirling numbers of the first kind to the bernoulli numbers with b1 12 are 1 m k 0 m 1 k m 1 k 1 b k 1 m 1 displaystyle frac 1msum k0m1kleftbkfrac 1m1 and the inversion of this sum for n 0 m 0 1 m k 0 m 1 k m 1 k 1 b n k a n m displaystyle frac 1msum k0m1kleftbnkanm here the number anm are the rational akiyamatanigawa numbers the first few of which are displayed in the following table akiyamatanigawa number mn 0 1 2 3 4 0 1 12 13 14 15 1 12 13 14 15 2 16 16 320 3 0 130 4 130 the akiyamatanigawa numbers satisfy a simple recurrence relation which can be exploited to iteratively compute the bernoulli numbers this leads to the algorithm shown in the section algorithmic description above see oeisa051714oeisa051715 an autosequence is a sequence which has its inverse binomial transform equal to the signed sequence if the main diagonal is zeroes oeisa000004 the autosequence is of the first kind example oeisa000045 the fibonacci numbers if the main diagonal is the first upper diagonal multiplied by 2 it is of the second kind example oeisa164555oeisa027642 the second bernoulli numbers see oeisa190339 the akiyamatanigawa transform applied to 2n 1oeisa000079 leads to oeisa198631 n oeisa06519 n 1 hence akiyamatanigawa transform for the second euler numbers mn 0 1 2 3 4 0 1 12 14 18 116 1 12 12 38 14 2 0 14 38 3 14 14 4 0 see oeisa209308 and oeisa227577 oeisa198631 n oeisa006519 n 1 are the second fractional euler numbers and an autosequence of the second kind oeisa164555 n 2oeisa027642 n 2 16 0 130 0 142 2n 3 2n 2 3 143 152 625 21 oeisa198631 n 1oeisa006519 n 2 12 0 14 0 12 also valuable for oeisa027641 oeisa027642 see connection with worpitzky numbers connection with pascals triangle there are formulas connecting pascals triangle to bernoulli numbers b n a n n 1 displaystyle bnfrac ann1 where a n displaystyle an is the determinant of a nbyn square matrix part of pascals triangle whose elements are a i k 0 if k 1 i i 1 k 1 otherwise displaystyle aikbegincases0textif k1ii1 choose k1textotherwiseendcases example b 6 det 1 2 0 0 0 0 1 3 3 0 0 0 1 4 6 4 0 0 1 5 10 10 5 0 1 6 15 20 15 6 1 7 21 35 35 21 7 120 5040 1 42 displaystyle b6frac det beginpmatrix120000133000146400151010501615201561721353521endpmatrix7frac 1205040frac 142 connection with eulerian numbers there are formulas connecting eulerian numbers nm to bernoulli numbers m 0 n 1 m n m 2 n 1 2 n 1 1 b n 1 n 1 m 0 n 1 m n m n m 1 n 1 b n displaystyle beginalignedsum m0n1mleftlangle n atop mrightrangle 2n12n11frac bn1n1sum m0n1mleftlangle n atop mrightrangle binom nm1n1bnendaligned both formulae are valid for n 0 if b1 is set to 12 if b1 is set to 12 they are valid only for n 1 and n 2 respectively a binary tree representation the stirling polynomials nx are related to the bernoulli numbers by bn nn1 s c woon woon 1997 described an algorithm to compute n1 as a binary tree woons recursive algorithm for n 1 starts by assigning to the root node n in the initial part of the tree represented above with denoting the sign of a1 given a node n the factorial of n is defined as n a 1 k 2 length n a k displaystyle na1prod k2operatorname length nak restricted to the nodes n of a fixed treelevel n the sum of 1n is n1 thus b n treelevel n n node of n n displaystyle bnsum stackrel ntext node oftext treelevel nfrac nn for example b1 112 b2 213 122 b3 314 123 132 1222 integral representation and continuation the integral b s 2 e s i 2 0 s t s 1 e 2 t d t t displaystyle bs2esipi 2int 0infty frac sts1e2pi tfrac dtt has as special values b2n b2n for n 0 for example b3 3233i and b5 15255i here is the riemann zeta function and i is the imaginary unit leonhard euler opera omnia ser 1 vol 10 p351 considered these numbers and calculated p 3 2 3 1 1 2 3 1 3 3 00581522 q 15 2 5 1 1 2 5 1 3 5 00254132 displaystyle beginalignedpfrac 32pi 3left1frac 123frac 133cdots right00581522ldots qfrac 152pi 5left1frac 125frac 135cdots right00254132ldots endaligned the relation to the euler numbers and the euler numbers are a sequence of integers intimately connected with the bernoulli numbers comparing the asymptotic expansions of the bernoulli and the euler numbers shows that the euler numbers e2n are in magnitude approximately 242n 22n times larger than the bernoulli numbers b2n in consequence 2 2 2 n 4 2 n b 2 n e 2 n displaystyle pi sim 222n42nfrac b2ne2n this asymptotic equation reveals that lies in the common root of both the bernoulli and the euler numbers in fact could be computed from these rational approximations bernoulli numbers can be expressed through the euler numbers and vice versa since for odd n bn en 0 with the exception b1 it suffices to consider the case when n is even b n k 0 n 1 n 1 k n 4 n 2 n e k n 2 4 6 e n k 1 n n k 1 2 k 4 k k b k n 2 4 6 displaystyle beginalignedbnsum k0n1binom n1kfrac n4n2nekn246ldots ensum k1nbinom nk1frac 2k4kkbkn246ldots endaligned these conversion formulas express an inverse relation between the bernoulli and the euler numbers but more important there is a deep arithmetic root common to both kinds of numbers which can be expressed through a more fundamental sequence of numbers also closely tied to these numbers are defined for n 1 as s n 2 2 n k 4 k 1 n k 0 1 1 2 2 displaystyle sn2leftfrac 2pi rightnsum kinfty infty 4k1nqquad k01122ldots and s1 1 by convention elkies 2003 the magic of these numbers lies in the fact that they turn out to be rational numbers this was first proved by leonhard euler in a landmark paper euler 1735 de summis serierum reciprocarum on the sums of series of reciprocals and has fascinated mathematicians ever since the first few of these numbers are s n 1 1 1 2 1 3 5 24 2 15 61 720 17 315 277 8064 62 2835 displaystyle sn11frac 12frac 13frac 524frac 215frac 61720frac 17315frac 2778064frac 622835ldots oeisa099612 oeisa099617 these are the coefficients in the expansion of sec x tan x the bernoulli numbers and euler numbers are best understood as special views of these numbers selected from the sequence sn and scaled for use in special applications b n 1 n 2 n even n 2 n 4 n s n n 2 3 e n 1 n 2 n even n s n 1 n 0 1 displaystyle beginalignedbn1leftlfloor frac n2rightrfloor nsn1n01ldots endaligned the expression has the value 1 if n is even and 0 otherwise iverson bracket these identities show that the quotient of bernoulli and euler numbers at the beginning of this section is just the special case of rn 2snsn 1 when n is even the rn are rational approximations to and two successive terms always enclose the true value of beginning with n 1 the sequence starts oeisa132049 oeisa132050 2 4 3 16 5 25 8 192 61 427 136 4352 1385 12465 3968 158720 50521 displaystyle 243frac 165frac 258frac 19261frac 427136frac 43521385frac 124653968frac 15872050521ldots quad longrightarrow pi these rational numbers also appear in the last paragraph of eulers paper cited above consider the akiyamatanigawa transform for the sequence oeisa046978 n 2 oeisa016116 n 1 0 1 12 0 14 14 18 0 1 12 1 34 0 58 34 2 12 12 94 52 58 3 1 72 34 152 4 52 112 994 5 8 772 6 612 from the second the numerators of the first column are the denominators of eulers formula the first column is 12 oeisa163982 an algorithmic view the seidel triangle the sequence sn has another unexpected yet important property the denominators of sn divide the factorial n1 in other words the numbers tnsnn1 sometimes called euler zigzag numbers are integers t n 1 1 1 2 5 16 61 272 1385 7936 50521 353792 n 0 1 2 3 displaystyle tn1112516612721385793650521353792ldots quad n0123ldots oeisa000111 see oeisa253671 thus the above representations of the bernoulli and euler numbers can be rewritten in terms of this sequence as b n 1 n 2 n even n 2 n 4 n t n 1 n 2 3 e n 1 n 2 n even t n 1 n 0 1 displaystyle beginalignedbn1leftlfloor frac n2rightrfloor tn1n01ldots endaligned these identities make it easy to compute the bernoulli and euler numbers the euler numbers en are given immediately by t2n 1 and the bernoulli numbers b2n are obtained from t2n by some easy shifting avoiding rational arithmetic what remains is to find a convenient way to compute the numbers tn however already in 1877 philipp ludwig von seidel seidel 1877 published an ingenious algorithm which makes it extremely simple to calculate tn seidels algorithm for tn start by putting 1 in row 0 and let k denote the number of the row currently being filled if k is odd then put the number on the left end of the row k 1 in the first position of the row k and fill the row from the left to the right with every entry being the sum of the number to the left and the number to the upper at the end of the row duplicate the last number if k is even proceed similar in the other direction seidels algorithm is in fact much more general see the exposition of dominique dumont dumont 1981 and was rediscovered several times thereafter similar to seidels approach d e knuth and t j buckholtz knuth buckholtz 1967 gave a recurrence equation for the numbers t2n and recommended this method for computing b2n and e2n on electronic computers using only simple operations on integers v i arnold rediscovered seidels algorithm in arnold 1991 and later millar sloane and young popularized seidels algorithm under the name boustrophedon transform triangular form 1 1 1 2 2 1 2 4 5 5 16 16 14 10 5 16 32 46 56 61 61 272 272 256 224 178 122 61 only oeisa000657 with one 1 and oeisa214267 with two 1s are in the oeis distribution with a supplementary 1 and one 0 in the following rows 1 0 1 1 1 0 0 1 2 2 5 5 4 2 0 0 5 10 14 16 16 61 61 56 46 32 16 0 this is oeisa239005 a signed version of oeisa008280 the main andiagonal is oeisa122045 the main diagonal is oeisa155585 the central column is oeisa099023 row sums 1 1 2 5 16 61 see oeisa163747 see the array beginning with 1 1 0 2 0 16 0 below the akiyamatanigawa algorithm applied to oeisa046978 n 1 oeisa016116n yields 1 1 12 0 14 14 18 0 1 32 1 0 34 1 1 32 4 154 0 5 152 1 5 5 512 0 61 61 1 the first column is oeisa122045 its binomial transform leads to 1 1 0 2 0 16 0 0 1 2 2 16 16 1 1 4 14 32 0 5 10 46 5 5 56 0 61 61 the first row of this array is oeisa155585 the absolute values of the increasing antidiagonals are oeisa008280 the sum of the antidiagonals is oeisa163747 n 1 2 the second column is 1 1 1 5 5 61 61 1385 1385 its binomial transform yields 1 2 2 4 16 32 272 1 0 6 12 48 240 1 6 6 60 192 5 0 66 32 5 66 66 61 0 61 the first row of this array is 1 2 2 4 16 32 272 544 7936 15872 353792 707584 the absolute values of the second bisection are the double of the absolute values of the first bisection consider the akiyamatanigawa algorithm applied to oeisa046978 n oeisa158780 n 1 absoeisa117575 n 1 1 2 2 32 1 34 34 78 1 1716 1716 3332 1 2 2 32 1 34 34 1 0 32 2 54 0 1 3 32 3 254 2 3 272 13 5 21 32 16 45 61 the first column whose the absolute values are oeisa000111 could be the numerator of a trigonometric function oeisa163747 is an autosequence of the first kind the main diagonal is oeisa000004 the corresponding array is 0 1 1 2 5 16 61 1 0 3 3 21 45 1 3 0 24 24 2 3 24 0 5 21 24 16 45 61 the first two upper diagonals are 1 3 24 402 1n 1oeisa002832 the sum of the antidiagonals is 0 2 0 10 2oeisa122045n1 oeisa163982 is an autosequence of the second kind like for instance oeisa164555 oeisa027642 hence the array 2 1 1 2 5 16 61 1 2 1 7 11 77 1 1 8 4 88 2 7 4 92 5 11 88 16 77 61 the main diagonal here 2 2 8 92 is the double of the first upper one here oeisa099023 the sum of the antidiagonals is 2 0 4 0 2oeisa155585n 1 note that oeisa163747oeisa163982 2oeisa122045 a combinatorial view alternating permutations alternating permutations around 1880 three years after the publication of seidels algorithm dsir andr proved a now classic result of combinatorial analysis andr 1879 andr 1881 looking at the first terms of the taylor expansion of the trigonometric functions tan x and sec x andr made a startling discovery tan x x 2 x 3 3 16 x 5 5 272 x 7 7 7936 x 9 9 sec x 1 x 2 2 5 x 4 4 61 x 6 6 1385 x 8 8 50521 x 10 10 displaystyle beginalignedtan xxfrac 2x33frac 16x55frac 272x77frac 7936x99cdots sec x1frac x22frac 5x44frac 61x66frac 1385x88frac 50521x1010cdots endaligned the coefficients are the euler numbers of odd and even index respectively in consequence the ordinary expansion of tan x sec x has as coefficients the rational numbers sn tan x sec x 1 x 1 2 x 2 1 3 x 3 5 24 x 4 2 15 x 5 61 720 x 6 displaystyle tan xsec x1xtfrac 12x2tfrac 13x3tfrac 524x4tfrac 215x5tfrac 61720x6cdots andr then succeeded by means of a recurrence argument to show that the alternating permutations of odd size are enumerated by the euler numbers of odd index also called tangent numbers and the alternating permutations of even size by the euler numbers of even index also called secant numbers related sequences the arithmetic mean of the first and the second bernoulli numbers are the associate bernoulli numbers b0 1 b1 0 b2 16 b3 0 b4 130 oeisa176327 oeisa027642 via the second row of its inverse akiyamatanigawa transform oeisa177427 they lead to balmer series oeisa061037 oeisa061038 the akiyamatanigawa algorithm applied to oeisa060819 n 4 oeisa145979 n leads to the bernoulli numbers oeisa027641 oeisa027642 oeisa164555 oeisa027642 or oeisa176327 oeisa176289 without b1 named intrinsic bernoulli numbers bin 1 56 34 710 23 16 16 320 215 542 0 130 120 235 584 130 130 3140 1105 0 0 142 128 4105 128 hence another link between the intrinsic bernoulli numbers and the balmer series via oeisa145979 n oeisa145979 n 2 0 2 1 6 is a permutation of the nonnegative numbers the terms of the first row are fn 12 1n 2 2 fn is an autosequence of the second kind 32 fn leads by its inverse binomial transform to 32 12 13 14 15 12 log 2 consider gn 12 1 n2 0 16 14 310 13 the akiyamatanagiwa transforms gives 0 16 14 310 13 514 16 16 320 215 542 328 0 130 120 235 584 584 130 130 3140 1105 0 1140 0 gn is an autosequence of the second kind euler oeisa198631 n oeisa006519 n 1 without the second term 12 are the fractional intrinsic euler numbers ein 1 0 14 0 12 0 178 0 the corresponding akiyama transform is 1 1 78 34 2132 0 14 38 38 516 14 14 0 14 2564 0 12 34 916 532 12 12 916 138 12564 the first line is eun eun preceded by a zero is an autosequence of the first kind it is linked to the oresme numbers the numerators of the second line are oeisa069834 preceded by 0 the difference table is 0 1 1 78 34 2132 1932 1 0 18 18 332 116 5128 1 18 0 132 132 3128 164 generalization to the oddindex bernoulli numbers the bernoulli numbers bn for n 0 1 2 3 are 1 12 16 356 130 25992 142 42716256 130 12465261632 566 5557314102256 6912730 3513594567100672 76 29904147151073709056 oeisa193472oeisa193473 this is ezn 1n4n 2n where ezn is the nth coefficient of sec t tan t oeisa000111oeisa000142 a companion to the bernoulli numbers see oeisa190339 the following fractional numbers are an autosequence of the first kind oeisa191754 oeisa192366 0 12 12 13 16 115 130 135 170 1105 1210 411155 412310 5895005 58910010 apply tn 1 k 2tn k 1 tnk to t0k oeisa191754 koeisa192366k 0 12 12 13 16 115 1 12 16 0 130 0 0 16 16 115 130 121 13 16 130 215 13210 221 0 730 730 1105 53210 13105 715 730 53210 52105 1210 92105 the rows are alternatively autosequences of the first and of the second kind the second row is oeisa164555oeisa027642 for the third row see oeisa051716 the first column is 0 1 0 13 0 715 0 3121 0 127105 0 51133 from mersenne primes see oeisa141459 for the second column see oeisa140252 consider the triangle oeisa097805 n 1 fiban 0 1 0 1 1 0 1 2 1 0 this is pascals triangle oeisa0007318 bordered by zeroes the antidiagonals sums are oeisa000045 the fibonacci numbers two elementary transforms yield the array aspec0 a companion to aspec in oeisa191302 0 1 1 1 1 0 1 2 3 4 0 1 3 6 10 0 1 4 10 20 0 1 5 15 35 multiplying the sbd array in oeisa191302 by aspec0 we have by row sums oeisa191754oeisa192366 0 12 12 0 12 16 12 26 0 12 36 115 12 46 315 0 12 56 615 4105 this triangle is unreduced arithmetical properties of the bernoulli numbers the bernoulli numbers can be expressed in terms of the riemann zeta function as bn n1 n for integers n 0 provided for n 0 and n 1 the expression n1 n is understood as the limiting value and the convention b1 12 is used this intimately relates them to the values of the zeta function at negative integers as such they could be expected to have and do have deep arithmetical properties for example the agohgiuga conjecture postulates that p is a prime number if and only if pbp 1 is congruent to 1 modulo p divisibility properties of the bernoulli numbers are related to the ideal class groups of cyclotomic fields by a theorem of kummer and its strengthening in the herbrandribet theorem and to class numbers of real quadratic fields by ankenyartinchowla the kummer theorems the bernoulli numbers are related to fermats last theorem flt by kummers theorem kummer 1850 which says if the odd prime p does not divide any of the numerators of the bernoulli numbers b2 b4 bp 3 then xp yp zp 0 has no solutions in nonzero integers prime numbers with this property are called regular primes another classical result of kummer kummer 1851 are the following congruences kummers congruence let p be an odd prime and b an even number such that p1 does not divide b then for any nonnegative integer k b k p 1 b k p 1 b b b b mod p displaystyle frac bkp1bkp1bequiv frac bbbpmod p a generalization of these congruences goes by the name of padic continuity padic continuity if b m and n are positive integers such that m and n are not divisible by p 1 and m n mod pb 1 p 1 then 1 p m 1 b m m 1 p n 1 b n n mod p b displaystyle 1pm1frac bmmequiv 1pn1frac bnnpmod pb since bn n1 n this can also be written 1 p u u 1 p v v mod p b displaystyle left1purightzeta uequiv left1pvrightzeta vpmod pb where u 1 m and v 1 n so that u and v are nonpositive and not congruent to 1 modulo p 1 this tells us that the riemann zeta function with 1 ps taken out of the euler product formula is continuous in the padic numbers on odd negative integers congruent modulo p 1 to a particular a 1 mod p 1 and so can be extended to a continuous function ps for all padic integers p the padic zeta function ramanujans congruences the following relations due to ramanujan provide a method for calculating bernoulli numbers that is more efficient than the one given by their original recursive definition m 3 m b m m 3 3 j 1 m 6 m 3 m 6 j b m 6 j if m 0 mod 6 m 3 3 j 1 m 2 6 m 3 m 6 j b m 6 j if m 2 mod 6 m 3 6 j 1 m 4 6 m 3 m 6 j b m 6 j if m 4 mod 6 displaystyle binom m3mbmbegincasesfrac m33sum limits j1frac m6binom m3m6jbm6jtextif mequiv 0pmod 6frac m33sum limits j1frac m26binom m3m6jbm6jtextif mequiv 2pmod 6frac m36sum limits j1frac m46binom m3m6jbm6jtextif mequiv 4pmod 6endcases von staudtclausen theorem von staudtclausen theorem the von staudtclausen theorem was given by karl georg christian von staudt von staudt 1840 and thomas clausen clausen 1840 independently in 1840 the theorem states that for every n 0 b 2 n p 1 2 n 1 p displaystyle b2nsum p1mid 2nfrac 1p is an integer the sum extends over all primes p for which p 1 divides 2n a consequence of this is that the denominator of b2n is given by the product of all primes p for which p 1 divides 2n in particular these denominators are squarefree and divisible by6 why do the odd bernoulli numbers vanish the sum k n i 0 n i k n k 2 displaystyle varphi knsum i0nikfrac nk2 can be evaluated for negative values of the index n doing so will show that it is an odd function for even values of k which implies that the sum has only terms of odd index this and the formula for the bernoulli sum imply that b2k 1 m is 0 for m even and 2k 1 m 1 and that the term for b1 is cancelled by the subtraction the von staudtclausen theorem combined with worpitzkys representation also gives a combinatorial answer to this question valid for n 1 from the von staudtclausen theorem it is known that for odd n 1 the number 2bn is an integer this seems trivial if one knows beforehand that the integer in question is zero however by applying worpitzkys representation one gets 2 b n m 0 n 1 m 2 m 1 m n 1 m 1 0 n 1 is odd displaystyle 2bnsum m0n1mfrac 2m1mleftn1 atop m1right0quad n1text is odd as a sum of integers which is not trivial here a combinatorial fact comes to surface which explains the vanishing of the bernoulli numbers at odd index let snm be the number of surjective maps from 1 2 n to 1 2 m then snm mnm the last equation can only hold if odd m 1 n 1 2 m 2 s n m even m 2 n 2 m 2 s n m n 2 is even displaystyle sum textodd m1n1frac 2m2snmsum texteven m2nfrac 2m2snmquad n2text is even this equation can be proved by induction the first two examples of this equation are n 4 2 8 7 3 n 6 2 120 144 31 195 40 thus the bernoulli numbers vanish at odd index because some nonobvious combinatorial identities are embodied in the bernoulli numbers a restatement of the riemann hypothesis the connection between the bernoulli numbers and the riemann zeta function is strong enough to provide an alternate formulation of the riemann hypothesis rh which uses only the bernoulli number in fact marcel riesz riesz 1916 proved that the rh is equivalent to the following assertion for every 14 there exists a constant c 0 depending on such that rx cx as x here rx is the riesz function r x 2 k 1 k k x k 2 2 k b 2 k 2 k 2 k 1 k k x k 2 2 k 2 k displaystyle rx2sum k1infty frac koverline kxk2pi 2kleftfrac b2k2kright2sum k1infty frac koverline kxk2pi 2kbeta 2k nk denotes the rising factorial power in the notation of d e knuth the numbers n bnn occur frequently in the study of the zeta function and are significant because n is a pinteger for primes p where p 1 does not divide n the n are called divided bernoulli numbers generalized bernoulli numbers the generalized bernoulli numbers are certain algebraic numbers defined similarly to the bernoulli numbers that are related to special values of dirichlet lfunctions in the same way that bernoulli numbers are related to special values of the riemann zeta function let be a dirichlet character modulo f the generalized bernoulli numbers attached to are defined by a 1 f a t e a t e f t 1 k 0 b k t k k displaystyle sum a1fchi afrac teateft1sum k0infty bkchi frac tkk apart from the exceptional b11 12 we have for any dirichlet character that bk 0 if 1 1k generalizing the relation between bernoulli numbers and values of the riemann zeta function at nonpositive integers one has the for all integers k 1 l 1 k b k k displaystyle l1kchi frac bkchi k where ls is the dirichlet lfunction of further information eisensteinkronecker number appendix assorted identities umbral calculus gives a compact form of bernoullis formula by using an abstract symbol b s m n 1 m 1 b n m 1 b m 1 displaystyle smnfrac 1m1mathbf b nm1bm1 where the symbol bk that appears during binomial expansion of the parenthesized term is to be replaced by the bernoulli number bk and b1 12 more suggestively and mnemonically this may be written as a definite integral s m n 0 n b x m d x displaystyle smnint 0nmathbf b xmdx many other bernoulli identities can be written compactly with this symbol eg b 1 m b m displaystyle mathbf b 1mbm let n be nonnegative and even n 1 n 2 1 b n 2 n 2 n displaystyle zeta nfrac 1frac n21bn2pi n2n the nth cumulant of the uniform probability distribution on the interval b n n 1 0 0 1 2 1 0 0 n n 1 1 0 n 1 n 2 0 n 1 0 0 1 1 2 1 0 0 1 n 1 n 1 1 0 1 n 1 1 n 1 2 0 displaystyle beginalignedbnnbeginvmatrix10cdots 0121cdots 00vdots vdots vdots vdots nn1cdots 10n1ncdots 20endvmatrixnbeginvmatrix10cdots 01frac 121cdots 00vdots vdots vdots vdots frac 1nfrac 1n1cdots 10frac 1n1frac 1ncdots frac 120endvmatrixendaligned thus the determinant is n1 the stirling polynomial at x 1for evennumbered bernoulli numbers b2p is given by the p 1 p 1 determinant b 2 p 2 p 2 2 p 2 1 0 0 0 1 1 3 1 0 0 0 1 5 1 3 1 0 0 1 2 p 1 1 2 p 1 1 2 p 3 1 3 0 displaystyle b2pfrac 2p22p2beginvmatrix100cdots 01frac 1310cdots 00frac 15frac 131cdots 00vdots vdots vdots vdots vdots frac 12p1frac 12p1frac 12p3cdots frac 130endvmatrix let n 1 then leonhard euler 1 n k 1 n n k b k b n k b n 1 b n displaystyle frac 1nsum k1nbinom nkbkbnkbn1bn let n 1 then von ettingshausen 1827 k 0 n n 1 k n k 1 b n k 0 displaystyle sum k0nbinom n1knk1bnk0 let n 0 then leopold kronecker 1883 b n k 1 n 1 1 k k n 1 k j 1 k j n displaystyle bnsum k1n1frac 1kkbinom n1ksum j1kjn let n 1 and m 1 then carlitz 1968 1 m r 0 m m r b n r 1 n s 0 n n s b m s displaystyle 1msum r0mbinom mrbnr1nsum s0nbinom nsbms let n 4 and h n k 1 n k 1 displaystyle hnsum k1nk1 the harmonic number then h miki 1978 n 2 k 2 n 2 b n k n k b k k k 2 n 2 n k b n k n k b k h n b n displaystyle frac n2sum k2n2frac bnknkfrac bkksum k2n2binom nkfrac bnknkbkhnbn let n 4 yuri matiyasevich found 1997 n 2 k 2 n 2 b k b n k 2 l 2 n 2 n 2 l b l b n l n n 1 b n displaystyle n2sum k2n2bkbnk2sum l2n2binom n2lblbnlnn1bn faberpandharipandezagiergessel identity for n 1 n 2 b n 1 x k 1 n 1 b k x k b n k x n k k 0 n 1 n k b n k n k b k x h n 1 b n x displaystyle frac n2leftbn1xsum k1n1frac bkxkfrac bnkxnkrightsum k0n1binom nkfrac bnknkbkxhn1bnx choosing x 0 or x 1 results in the bernoulli number identity in one or another conventionthe next formula is true for n 0 if b1 b11 12 but only for n 1 if b1 b10 12 k 0 n n k b k n k 2 b n 1 n 1 displaystyle sum k0nbinom nkfrac bknk2frac bn1n1 let n 0 then 1 k 0 n n k 2 n k 1 n k 1 b k 1 2 n displaystyle 1sum k0nbinom nkfrac 2nk1nk1bk12n and 1 k 0 n n k 2 n k 1 n k 1 b k 0 n 0 displaystyle 1sum k0nbinom nkfrac 2nk1nk1bk0delta n0 a reciprocity relation of mbgelfand agoh dilcher 2008 1 m 1 j 0 k k j b m 1 j m 1 j 1 k 1 j 0 m m j b k 1 j k 1 j k m k m 1 displaystyle 1m1sum j0kbinom kjfrac bm1jm1j1k1sum j0mbinom mjfrac bk1jk1jfrac kmkm1 bernoulli polynomial bernoulli polynomials of the second kind euler number genocchi number kummers congruences polybernoulli number hurwitz zeta function euler summation stirling polynomial sums of powers notes abramowitz m stegun c a 1972 231 bernoulli and euler polynomials and the eulermaclaurin formula handbook of mathematical functions with formulas graphs and mathematical tables 9th printing ed new york dover pp804806 agoh takashi dilcher karl 2008 reciprocity relations for bernoulli numbers american mathematical monthly 115 237244 jstor27642447 andr d 1879 dveloppements de sec x et tan x comptes rendus acad sci 88 965967 andr d 1881 mmoire sur les permutations alternes journal de mathmatiques pures et appliques 7 167184 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