Inmathematics, theBernoulli numbersB_n are asequence ofrational numbers which occur frequently inanalysis. The Bernoulli numbers appear in (and can be defined by) theTaylor series expansions of thetangent andhyperbolic tangent functions, inFaulhaber's formula for the sum ofm-th powers of the firstn positive integers, in theEuler–Maclaurin formula, and in expressions for certain values of theRiemann zeta function.

The values of the first 20 Bernoulli numbers are given in the adjacent table. Two conventions are used in the literature, denoted here by${\displaystyle B_n^{-}}$ and${\displaystyle B_n^{+}}$; they differ only forn = 1, where${\displaystyle B_1^{-}=-1/2}$ and${\displaystyle B_1^{+}=+1/2}$. For every oddn > 1,B_n = 0. For every evenn > 0,B_n is negative ifn is divisible by 4 and positive otherwise. The Bernoulli numbers are special values of theBernoulli polynomials${\displaystyle B_n(x)}$, with${\displaystyle B_n^{-}=B_n(0)}$ and${\displaystyle B_n^{+}=B_n(1)}$.^[1]

The Bernoulli numbers were discovered around the same time by the Swiss mathematicianJacob Bernoulli, after whom they are named, and independently by Japanese mathematicianSeki Takakazu. Seki's discovery was posthumously published in 1712^[2][3][4] in his workKatsuyō Sanpō; Bernoulli's, also posthumously, in hisArs Conjectandi of 1713.Ada Lovelace'snote G on theAnalytical Engine from 1842 describes analgorithm for generating Bernoulli numbers withBabbage's machine;^[5] it is disputedwhether Lovelace or Babbage developed the algorithm. As a result, the Bernoulli numbers have the distinction of being the subject of the first published complexcomputer program.

The superscript± used in this article distinguishes the two sign conventions for Bernoulli numbers. Only then = 1 term is affected:

• B⁻
  _n withB⁻
  ₁ = −⁠1/2⁠ (OEIS: A027641 /OEIS: A027642) is the sign convention prescribed byNIST and most modern textbooks.^[6]
• B⁺
  _n withB⁺
  ₁ = +⁠1/2⁠ (OEIS: A164555 /OEIS: A027642) was used in the older literature,^[1] and (since 2022) byDonald Knuth^[7] following Peter Luschny's "Bernoulli Manifesto".^[8]

In the formulas below, one can switch from one sign convention to the other with the relation${\displaystyle B_n^{+}=(-1{)}^n{B}_n^{-}}$ , or for integern = 2 or greater, simply ignore it.

SinceB_n = 0 for all oddn > 1, and many formulas only involve even-index Bernoulli numbers, a few authors write "B_n" instead ofB_2n . This article does not follow that notation.

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.

Methods to calculate the sum of the firstn positive integers, the sum of the squares and of the cubes of the firstn 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 werePythagoras (c. 572–497 BCE, Greece),Archimedes (287–212 BCE, Italy),Aryabhata (b. 476, India),Abu Bakr al-Karaji (d. 1019, Persia) and Abu Ali al-Hasan ibn al-Hasan ibnal-Haytham (965–1039, Iraq).

During the late sixteenth and early seventeenth centuries mathematicians made significant progress. In the WestThomas Harriot (1560–1621) of England,Johann Faulhaber (1580–1635) of Germany,Pierre de Fermat (1601–1665) and fellow French mathematicianBlaise Pascal (1623–1662) 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 1631Academia Algebrae, far higher than anyone before him, but he did not give a general formula.

Blaise Pascal in 1654 provedPascal's identity relating(n+1)^k+1 to the sums of thepth powers of the firstn positive integers forp = 0, 1, 2, ...,k.

The Swiss mathematician Jakob Bernoulli (1654–1705) was the first to realize the existence of a single sequence of constantsB₀,B₁,B₂,... which provide a uniform formula for all sums of powers.^[9]

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 thecth powers for any positive integerc 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 91,409,924,241,424,243,424,241,924,242,500."

Bernoulli's result was published posthumously inArs Conjectandi in 1713.Seki Takakazu independently discovered the Bernoulli numbers and his result was published a year earlier, also posthumously, in 1712.^[2] However, Seki did not present his method as a formula based on a sequence of constants.

Bernoulli's formula for sums of powers is the most useful and generalizable formulation to date. The coefficients in Bernoulli's formula are now called Bernoulli numbers, following a suggestion ofAbraham de Moivre.

Bernoulli's formula is sometimes calledFaulhaber's formula after Johann Faulhaber who found remarkable ways to calculate sum of powers but never stated Bernoulli's formula. According to Knuth^[9] a rigorous proof of Faulhaber's formula was first published byCarl Jacobi in 1834.^[10] Knuth's in-depth study of Faulhaber's formula concludes (the nonstandard notation on the LHS is explained further on):

"Faulhaber never discovered the Bernoulli numbers; i.e., he never realized that a single sequence of constantsB₀,B₁,B₂,... would provide a uniform

$\sum n^m=\frac{1}{m+1}\left(B_0{n}^{m+1}-(\genfrac{}{}{0ex}{}{m+1}{1})B_1{n}^m+(\genfrac{}{}{0ex}{}{m+1}{2})B_2{n}^{m-1}-\cdots +(-1{)}^m(\genfrac{}{}{0ex}{}{m+1}{m})B_{m}n\right)$ 

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 forΣn^mfrom polynomials inN to polynomials inn."^[11]

In the above Knuth meant${\displaystyle B_1^{-}}$ ; instead using${\displaystyle B_1^{+}}$  the formula avoids subtraction:

$\sum n^m=\frac{1}{m+1}\left(B_0{n}^{m+1}+(\genfrac{}{}{0ex}{}{m+1}{1})B_1^{+}{n}^m+(\genfrac{}{}{0ex}{}{m+1}{2})B_2{n}^{m-1}+\cdots +(\genfrac{}{}{0ex}{}{m+1}{m})B_{m}n\right).$ 

The Bernoulli numbersOEIS: A164555(n)/OEIS: A027642(n) were introduced by Jakob Bernoulli in the bookArs Conjectandi published posthumously in 1713 page 97. The main formula can be seen in the second half of the corresponding facsimile. The constant coefficients denotedA,B,C andD by Bernoulli are mapped to the notation which is now prevalent asA =B₂,B =B₄,C =B₆,D =B₈. The expressionc·c−1·c−2·c−3 meansc·(c−1)·(c−2)·(c−3) – the small dots are used as grouping symbols. Using today's terminology these expressions arefalling factorial powersc^k. The factorial notationk! as a shortcut for1 × 2 × ... ×k was not introduced until 100 years later. The integral symbol on the left hand side goes back toGottfried Wilhelm Leibniz in 1675 who used it as a long letterS for "summa" (sum).^[b] The lettern on the left hand side is not an index ofsummation but gives the upper limit of the range of summation which is to be understood as1, 2, ...,n. Putting things together, for positivec, today a mathematician is likely to write Bernoulli's formula as:

${\displaystyle \sum _{k=1}^n{k}^c=\frac{n^{c+1}}{c+1}+\frac{1}{2}{n}^c+\sum _{k=2}^c\frac{B_k}{k!}{c}^{\underset{\_}{k-1}}{n}^{c-k+1}.}$ 

This formula suggests settingB₁ =⁠1/2⁠ when switching from the so-called '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 thefalling factorialc^k−1 has fork = 0 the value⁠1/c + 1⁠.^[12] Thus Bernoulli's formula can be written

${\displaystyle \sum _{k=1}^n{k}^c=\sum _{k=0}^c\frac{B_k}{k!}{c}^{\underset{\_}{k-1}}{n}^{c-k+1}}$ 

ifB₁ = 1/2, recapturing the value Bernoulli gave to the coefficient at that position.

The formula for${\displaystyle \sum _{k=1}^n{k}^9}$  in the first half of the quotation by Bernoulli above contains an error at the last term; it should be${\displaystyle -\frac{3}{20}{n}^2}$  instead of${\displaystyle -\frac{1}{12}{n}^2}$ .

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 four of the most useful ones are mentioned:

• a recursive equation,
• an explicit formula,
• a generating function,
• an integral expression.

For the proof of theequivalence of the four approaches.^[13]

The Bernoulli numbers obey the sum formulas^[1]

 

where${\displaystyle m=0,1,\mathrm{2...}}$  andδ denotes theKronecker delta.

The first of these is sometimes written^[14] as the formula (for m > 1)$${\displaystyle (B+1{)}^m-B_m=0,}$$ where the power is expanded formally using the binomial theorem and${\displaystyle B^k}$  is replaced by${\displaystyle B_k}$ .

Solving for${\displaystyle B_m^{\mp }}$  gives the recursive formulas^[15]

 

In 1893Louis Saalschütz listed a total of 38 explicit formulas for the Bernoulli numbers,^[16] usually giving some reference in the older literature. One of them is (for${\displaystyle m\ge 1}$ ):

 

The exponentialgenerating functions are

 

where the substitution is${\displaystyle t\to -t}$ . The two generating functions only differ byt.

The (ordinary) generating function

${\displaystyle z^{-1}{\psi }_1(z^{-1})=\sum _{m=0}^{\mathrm{\infty }}{B}_m^{+}{z}^m}$ 

is anasymptotic series. It contains thetrigamma functionψ₁.

From the generating functions above, one can obtain the following integral formula for the even Bernoulli numbers:

${\displaystyle B_{2n}=4n(-1{)}^{n+1}{\int }_0^{\mathrm{\infty }}\frac{t^{2n-1}}{e^{2\pi t}-1}\mathrm{d}t}$ 

The Bernoulli numbers can be expressed in terms of theRiemann zeta function:

B⁺
_n = −n ζ(1 −n)           forn ≥ 1 .

Here the argument of the zeta function is0or negative. As${\displaystyle \zeta (k)}$  is zero for negative even integers (thetrivial zeroes), ifn>1 is odd,${\displaystyle \zeta (1-n)}$  is zero.

By means of the zetafunctional equation and the gammareflection formula the following relation can be obtained:^[17]

${\displaystyle B_{2n}=\frac{(-1{)}^{n+1}2(2n)!}{(2\pi {)}^{2n}}\zeta (2n)}$  forn ≥ 1 .

Now the argument of the zeta function is positive.

It then follows fromζ → 1 (n → ∞) andStirling's formula that

${\displaystyle |B_{2n}|\sim 4\sqrt{\pi n}{\left(\frac{n}{\pi e}\right)}^{2n}}$  forn → ∞ .

In some applications it is useful to be able to compute the Bernoulli numbersB₀ throughB_{p − 3} modulop, wherep is a prime; for example to test whetherVandiver's conjecture holds forp, or even just to determine whetherp is anirregular 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^[18] which require onlyO(p (logp)²) operations (seebigO notation).

David Harvey^[19] describes an algorithm for computing Bernoulli numbers by computingB_n modulop for many small primesp, and then reconstructingB_n via theChinese remainder theorem. Harvey writes that theasymptotictime complexity of this algorithm isO(n² log(n)^{2 +ε}) and claims that thisimplementation is significantly faster than implementations based on other methods. Using this implementation Harvey computedB_n forn = 10⁸. Harvey's implementation has been included inSageMath since version 3.1. Prior to that, Bernd Kellner^[20] computedB_n to full precision forn = 10⁶ in December 2002 and Oleksandr Pavlyk^[21] forn = 10⁷ withMathematica 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	1672	3330
G. Fee,S. Plouffe	1996	10000	27677
G. Fee, S. Plouffe	1996	100000	376755
B. C. Kellner	2002	1000000	4767529
O. Pavlyk	2008	10000000	57675260
D. Harvey	2008	100000000	676752569

*Digits is to be understood as the exponent of 10 whenB_n is written as a real number in normalizedscientific notation.

Arguably the most important application of the Bernoulli numbers in mathematics is their use in theEuler–Maclaurin formula. Assuming thatf is a sufficiently often differentiable function the Euler–Maclaurin formula can be written as^[22]

${\displaystyle \sum _{k=a}^{b-1}f(k)={\int }_a^{b}f(x)dx+\sum _{k=1}^m\frac{B_k^{-}}{k!}(f^{(k-1)}(b)-f^{(k-1)}(a))+R_{-}(f,m).}$ 

This formulation assumes the conventionB⁻
₁ = −⁠1/2⁠. Using the conventionB⁺
₁ = +⁠1/2⁠ the formula becomes

${\displaystyle \sum _{k=a+1}^{b}f(k)={\int }_a^{b}f(x)dx+\sum _{k=1}^m\frac{B_k^{+}}{k!}(f^{(k-1)}(b)-f^{(k-1)}(a))+R_{+}(f,m).}$ 

Here${\displaystyle f^{(0)}=f}$  (i.e. the zeroth-order derivative of${\displaystyle f}$  is just${\displaystyle f}$ ). Moreover, let${\displaystyle f^{(-1)}}$  denote anantiderivative of${\displaystyle f}$ . By thefundamental theorem of calculus,

${\displaystyle {\int }_a^{b}f(x)dx=f^{(-1)}(b)-f^{(-1)}(a).}$ 

Thus the last formula can be further simplified to the following succinct form of the Euler–Maclaurin formula

${\displaystyle \sum _{k=a+1}^{b}f(k)=\sum _{k=0}^m\frac{B_k}{k!}(f^{(k-1)}(b)-f^{(k-1)}(a))+R(f,m).}$ 

This form is for example the source for the important Euler–Maclaurin expansion of the zeta function

 

Heres^k denotes therising factorial power.^[23]

Bernoulli numbers are also frequently used in other kinds ofasymptotic expansions. The following example is the classical Poincaré-type asymptotic expansion of thedigamma functionψ.

${\displaystyle \psi (z)\sim \ln z-\sum _{k=1}^{\mathrm{\infty }}\frac{B_k^{+}}{k{z}^k}}$ 

Bernoulli numbers feature prominently in theclosed form expression of the sum of themth powers of the firstn positive integers. Form,n ≥ 0 define

${\displaystyle S_m(n)=\sum _{k=1}^n{k}^m=1^m+2^m+\cdots +n^m.}$ 

This expression can always be rewritten as apolynomial inn of degreem + 1. Thecoefficients of these polynomials are related to the Bernoulli numbers byBernoulli's formula:

${\displaystyle S_m(n)=\frac{1}{m+1}\sum _{k=0}^m(\genfrac{}{}{0ex}{}{m+1}{k})B_k^{+}{n}^{m+1-k}=m!\sum _{k=0}^m\frac{B_k^{+}{n}^{m+1-k}}{k!(m+1-k)!},}$ 

where(^{m + 1}
_k) denotes thebinomial coefficient.

For example, takingm to be 1 gives thetriangular numbers0, 1, 3, 6, ...OEIS: A000217.

${\displaystyle 1+2+\cdots +n=\frac{1}{2}(B_0{n}^2+2B_1^{+}{n}^1)=\frac{1}{2}(n^2+n).}$ 

Takingm to be 2 gives thesquare pyramidal numbers0, 1, 5, 14, ...OEIS: A000330.

${\displaystyle 1^2+2^2+\cdots +n^2=\frac{1}{3}(B_0{n}^3+3B_1^{+}{n}^2+3B_2{n}^1)=\frac{1}{3}\left(n^3+\frac{3}{2}{n}^2+\frac{1}{2}n\right).}$ 

Some authors use the alternate convention for Bernoulli numbers and state Bernoulli's formula in this way:

${\displaystyle S_m(n)=\frac{1}{m+1}\sum _{k=0}^m(-1{)}^k(\genfrac{}{}{0ex}{}{m+1}{k})B_k^{-}{n}^{m+1-k}.}$ 

Bernoulli's formula is sometimes calledFaulhaber's formula afterJohann Faulhaber who also found remarkable ways to calculatesums of powers.

Faulhaber's formula was generalized by V. Guo and J. Zeng to aq-analog.^[24]

The Bernoulli numbers appear in theTaylor series expansion of manytrigonometric functions andhyperbolic functions.

 

The Bernoulli numbers appear in the followingLaurent series:^[25]

Digamma function:${\displaystyle \psi (z)=\ln z-\sum _{k=1}^{\mathrm{\infty }}\frac{B_k^{+}}{k{z}^k}}$ 

TheKervaire–Milnor formula for the order of the cyclic group of diffeomorphism classes ofexotic(4n − 1)-spheres which boundparallelizable manifolds involves Bernoulli numbers. LetES_n be the number of such exotic spheres forn ≥ 2, then

${\displaystyle {\mathit{\text{ES}}}_n=(2^{2n-2}-2^{4n-3})\mathrm{Numerator}\left(\frac{B_{4n}}{4n}\right).}$ 

TheHirzebruch signature theorem for theL genus of asmoothorientedclosed manifold ofdimension 4n also involves Bernoulli 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, theinclusion–exclusion principle.

The definition to proceed with was developed by Julius Worpitzky in 1883. Besides elementary arithmetic only the factorial functionn! and the power functionk^m is employed. The signless Worpitzky numbers are defined as

${\displaystyle W_{n,k}=\sum _{v=0}^k(-1{)}^{v+k}(v+1{)}^n\frac{k!}{v!(k-v)!}.}$ 

They can also be expressed through theStirling numbers of the second kind

${\displaystyle W_{n,k}=k!\left\{\genfrac{}{}{0ex}{}{n+1}{k+1}\right\}.}$ 

A Bernoulli number is then introduced as an inclusion–exclusion sum of Worpitzky numbers weighted by theharmonic sequence 1, ⁠1/2⁠, ⁠1/3⁠, ...

${\displaystyle B_n=\sum _{k=0}^n(-1{)}^k\frac{W_{n,k}}{k+1}=\sum _{k=0}^n\frac{1}{k+1}\sum _{v=0}^k(-1{)}^v(v+1{)}^n(\genfrac{}{}{0ex}{}{k}{v}).}$ 

B₀ = 1

B₁ = 1 −⁠1/2⁠

B₂ = 1 −⁠3/2⁠ +⁠2/3⁠

B₃ = 1 −⁠7/2⁠ +⁠12/3⁠ −⁠6/4⁠

B₄ = 1 −⁠15/2⁠ +⁠50/3⁠ −⁠60/4⁠ +⁠24/5⁠

B₅ = 1 −⁠31/2⁠ +⁠180/3⁠ −⁠390/4⁠ +⁠360/5⁠ −⁠120/6⁠

B₆ = 1 −⁠63/2⁠ +⁠602/3⁠ −⁠2100/4⁠ +⁠3360/5⁠ −⁠2520/6⁠ +⁠720/7⁠

This representation hasB⁺
₁ = +⁠1/2⁠.

Consider the sequences_n,n ≥ 0. From Worpitzky's numbersOEIS: A028246,OEIS: A163626 applied tos₀,s₀,s₁,s₀,s₁,s₂,s₀,s₁,s₂,s₃, ... is identical to the Akiyama–Tanigawa transform applied tos_n (seeConnection with Stirling numbers of the first kind). This can be seen via the table:

Identity of
Worpitzky's representation and Akiyama–Tanigawa 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 representss₀,s₁,s₂,s₃,s₄.

Hence for the second fractional Euler numbersOEIS: A198631 (n) /OEIS: A006519 (n + 1):

E₀ = 1

E₁ = 1 −⁠1/2⁠

E₂ = 1 −⁠3/2⁠ +⁠2/4⁠

E₃ = 1 −⁠7/2⁠ +⁠12/4⁠ −⁠6/8⁠

E₄ = 1 −⁠15/2⁠ +⁠50/4⁠ −⁠60/8⁠ +⁠24/16⁠

E₅ = 1 −⁠31/2⁠ +⁠180/4⁠ −⁠390/8⁠ +⁠360/16⁠ −⁠120/32⁠

E₆ = 1 −⁠63/2⁠ +⁠602/4⁠ −⁠2100/8⁠ +⁠3360/16⁠ −⁠2520/32⁠ +⁠720/64⁠

A second formula representing the Bernoulli numbers by the Worpitzky numbers is forn ≥ 1

${\displaystyle B_n=\frac{n}{2^{n+1}-2}\sum _{k=0}^{n-1}(-2{)}^{-k}{W}_{n-1,k}.}$ 

The simplified second Worpitzky's representation of the second Bernoulli numbers is:

OEIS: A164555 (n + 1) /OEIS: A027642(n + 1) =⁠n + 1/2^{n + 2} − 2⁠ ×OEIS: A198631(n) /OEIS: A006519(n + 1)

which links the second Bernoulli numbers to the second fractional Euler numbers. The beginning is:

⁠1/2⁠,⁠1/6⁠, 0, −⁠1/30⁠, 0,⁠1/42⁠, ... = (⁠1/2⁠,⁠1/3⁠,⁠3/14⁠,⁠2/15⁠,⁠5/62⁠,⁠1/21⁠, ...) × (1,⁠1/2⁠, 0, −⁠1/4⁠, 0,⁠1/2⁠, ...)

The numerators of the first parentheses areOEIS: A111701 (seeConnection with Stirling numbers of the first kind).

If one defines theBernoulli polynomialsB_k(j) as:^[26]

${\displaystyle B_k(j)=k\sum _{m=0}^{k-1}(\genfrac{}{}{0ex}{}{j}{m+1})S(k-1,m)m!+B_k}$ 

whereB_k fork = 0, 1, 2,... are the Bernoulli numbers,andS(k,m) is aStirling number of the second kind.

One also has the following for Bernoulli polynomials,^[26]

${\displaystyle B_k(j)=\sum _{n=0}^k(\genfrac{}{}{0ex}{}{k}{n})B_n{j}^{k-n}.}$ 

The coefficient ofj in(^j
_{m + 1}) is⁠(−1)^m/m + 1⁠.

Comparing the coefficient ofj in the two expressions of Bernoulli polynomials, one has:

${\displaystyle B_k=\sum _{m=0}^{k-1}(-1{)}^m\frac{m!}{m+1}S(k-1,m)}$ 

(resulting inB₁ = +⁠1/2⁠) which is an explicit formula for Bernoulli numbers and can be used to proveVon-Staudt Clausen theorem.^[27][28][29]

The two main formulas relating the unsignedStirling numbers of the first kind[ⁿ
_m] to the Bernoulli numbers (withB₁ = +⁠1/2⁠) are

${\displaystyle \frac{1}{m!}\sum _{k=0}^m(-1{)}^k\left[\genfrac{}{}{0ex}{}{m+1}{k+1}\right]B_k=\frac{1}{m+1},}$ 

and the inversion of this sum (forn ≥ 0,m ≥ 0)

${\displaystyle \frac{1}{m!}\sum _{k=0}^m(-1{)}^k\left[\genfrac{}{}{0ex}{}{m+1}{k+1}\right]B_{n+k}=A_{n,m}.}$ 

Here the numberA_n,m are the rational Akiyama–Tanigawa numbers, the first few of which are displayed in the following table.

Akiyama–Tanigawa number

m n	0	1	2	3	4
0	1	⁠1/2⁠	⁠1/3⁠	⁠1/4⁠	⁠1/5⁠
1	⁠1/2⁠	⁠1/3⁠	⁠1/4⁠	⁠1/5⁠	...
2	⁠1/6⁠	⁠1/6⁠	⁠3/20⁠	...	...
3	0	⁠1/30⁠	...	...	...
4	−⁠1/30⁠	...	...	...	...

The Akiyama–Tanigawa 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. SeeOEIS: A051714/OEIS: A051715.

Anautosequence is a sequence which has its inverse binomial transform equal to the signed sequence. If the main diagonal is zeroes =OEIS: A000004, the autosequence is of the first kind. Example:OEIS: A000045, the Fibonacci numbers. If the main diagonal is the first upper diagonal multiplied by 2, it is of the second kind. Example:OEIS: A164555/OEIS: A027642, the second Bernoulli numbers (seeOEIS: A190339). The Akiyama–Tanigawa transform applied to2⁻ⁿ = 1/OEIS: A000079 leads toOEIS: A198631 (n) /OEIS: A06519 (n + 1). Hence:

Akiyama–Tanigawa transform for the second Euler numbers

m n	0	1	2	3	4
0	1	⁠1/2⁠	⁠1/4⁠	⁠1/8⁠	⁠1/16⁠
1	⁠1/2⁠	⁠1/2⁠	⁠3/8⁠	⁠1/4⁠	...
2	0	⁠1/4⁠	⁠3/8⁠	...	...
3	−⁠1/4⁠	−⁠1/4⁠	...	...	...
4	0	...	...	...	...

SeeOEIS: A209308 andOEIS: A227577.OEIS: A198631 (n) /OEIS: A006519 (n + 1) are the second (fractional) Euler numbers and an autosequence of the second kind.

(⁠OEIS: A164555 (n + 2)/OEIS: A027642 (n + 2)⁠ =⁠1/6⁠, 0, −⁠1/30⁠, 0,⁠1/42⁠, ...) × (⁠2^{n + 3} − 2/n + 2⁠ =3,⁠14/3⁠,⁠15/2⁠,⁠62/5⁠, 21, ...) =⁠OEIS: A198631 (n + 1)/OEIS: A006519 (n + 2)⁠ =⁠1/2⁠, 0, −⁠1/4⁠, 0,⁠1/2⁠, ....

Also valuable forOEIS: A027641 /OEIS: A027642 (seeConnection with Worpitzky numbers).

There are formulas connecting Pascal's triangle to Bernoulli numbers^[c]

${\displaystyle B_n^{+}=\frac{|A_n|}{(n+1)!}}$ 

where${\displaystyle |A_n|}$  is the determinant of a n-by-nHessenberg matrix part ofPascal's triangle whose elements are: 

Example:

 

There are formulas connectingEulerian numbers⟨ⁿ
_m⟩ to Bernoulli numbers:

 

Both formulae are valid forn ≥ 0 ifB₁ is set to⁠1/2⁠. IfB₁ is set to −⁠1/2⁠ they are valid only forn ≥ 1 andn ≥ 2 respectively.

The Stirling polynomialsσ_n(x) are related to the Bernoulli numbers byB_n =n!σ_n(1). S. C. Woon described an algorithm to computeσ_n(1) as a binary tree:^[30]

Woon's recursive algorithm (forn ≥ 1) starts by assigning to the root nodeN = [1,2]. Given a nodeN = [a₁,a₂, ...,a_k] of the tree, the left child of the node isL(N) = [−a₁,a₂ + 1,a₃, ...,a_k] and the right childR(N) = [a₁, 2,a₂, ...,a_k]. A nodeN = [a₁,a₂, ...,a_k] is written as±[a₂, ...,a_k] in the initial part of the tree represented above with ± denoting the sign ofa₁.

Given a nodeN the factorial ofN is defined as

${\displaystyle N!=a_1\prod _{k=2}^{\mathrm{length}(N)}{a}_k!.}$ 

Restricted to the nodesN of a fixed tree-leveln the sum of⁠1/N!⁠ isσ_n(1), thus

${\displaystyle B_n=\sum _{\stackrel{N\text{ node of}}{\text{ tree-level }n}}\frac{n!}{N!}.}$ 

For example:

B₁ = 1!(⁠1/2!⁠)

B₂ = 2!(−⁠1/3!⁠ +⁠1/2!2!⁠)

B₃ = 3!(⁠1/4!⁠ −⁠1/2!3!⁠ −⁠1/3!2!⁠ +⁠1/2!2!2!⁠)

Theintegral

${\displaystyle b(s)=2e^{si\pi /2}{\int }_0^{\mathrm{\infty }}\frac{s{t}^s}{1-e^{2\pi t}}\frac{dt}{t}=\frac{s!}{2^{s-1}}\frac{\zeta (s)}{{\pi }^s}(-i{)}^s=\frac{2s!\zeta (s)}{(2\pi i{)}^s}}$ 

has as special valuesb(2n) =B_2n forn > 0.

For example,b(3) =⁠3/2⁠ζ(3)π⁻³i andb(5) = −⁠15/2⁠ζ(5)π⁻⁵i. Here,ζ is theRiemann zeta function, andi is theimaginary unit. Leonhard Euler (Opera Omnia, Ser. 1, Vol. 10, p. 351) considered these numbers and calculated

 

Another similarintegral representation is

${\displaystyle b(s)=-\frac{e^{si\pi /2}}{2^s-1}{\int }_0^{\mathrm{\infty }}\frac{s{t}^s}{\sinh \pi t}\frac{dt}{t}=\frac{2e^{si\pi /2}}{2^s-1}{\int }_0^{\mathrm{\infty }}\frac{e^{\pi t}s{t}^s}{1-e^{2\pi t}}\frac{dt}{t}.}$ 

TheEuler numbers are a sequence of integers intimately connected with the Bernoulli numbers. Comparing theasymptotic expansions of the Bernoulli and the Euler numbers shows that the Euler numbersE_2n are in magnitude approximately⁠2/π⁠(4²ⁿ − 2²ⁿ) times larger than the Bernoulli numbersB_2n. In consequence:

${\displaystyle \pi \sim 2(2^{2n}-4^{2n})\frac{B_{2n}}{E_{2n}}.}$ 

This asymptotic equation reveals thatπ lies in the common root of both the Bernoulli and the Euler numbers. In factπ could be computed from these rational approximations.

Bernoulli numbers can be expressed through the Euler numbers and vice versa. Since, for oddn,B_n =E_n = 0 (with the exceptionB₁), it suffices to consider the case whenn is even.

 

These conversion formulas express a connection 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 toπ. These numbers are defined forn ≥ 1 as^[31][32]

${\displaystyle S_n=2{\left(\frac{2}{\pi }\right)}^n\sum _{k=0}^{\mathrm{\infty }}\frac{(-1{)}^{kn}}{(2k+1{)}^n}=2{\left(\frac{2}{\pi }\right)}^n\underset{K\to \mathrm{\infty }}{lim}\sum _{k=-K}^K(4k+1{)}^{-n}.}$ 

The magic of these numbers lies in the fact that they turn out to be rational numbers. This was first proved byLeonhard Euler in a landmark paperDe summis serierum reciprocarum (On the sums of series of reciprocals) and has fascinated mathematicians ever since.^[33] The first few of these numbers are

${\displaystyle S_n=1,1,\frac{1}{2},\frac{1}{3},\frac{5}{24},\frac{2}{15},\frac{61}{720},\frac{17}{315},\frac{277}{8064},\frac{62}{2835},\dots }$  (OEIS: A099612 /OEIS: A099617)

These are the coefficients in the expansion ofsecx + tanx.

The Bernoulli numbers and Euler numbers can be understood asspecial views of these numbers, selected from the sequenceS_n and scaled for use in special applications.

 

The expression [n even] has the value 1 ifn 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 ofR_n =⁠2S_n/S_{n + 1}⁠ whenn is even. TheR_n are rational approximations toπ and two successive terms always enclose the true value ofπ. Beginning withn = 1 the sequence starts (OEIS: A132049 /OEIS: A132050):

${\displaystyle 2,4,3,\frac{16}{5},\frac{25}{8},\frac{192}{61},\frac{427}{136},\frac{4352}{1385},\frac{12465}{3968},\frac{158720}{50521},\dots ⟶\pi .}$ 

These rational numbers also appear in the last paragraph of Euler's paper cited above.

Consider the Akiyama–Tanigawa transform for the sequenceOEIS: A046978 (n + 2) /OEIS: A016116 (n + 1):

0	1	⁠1/2⁠	0	−⁠1/4⁠	−⁠1/4⁠	−⁠1/8⁠	0
1	⁠1/2⁠	1	⁠3/4⁠	0	−⁠5/8⁠	−⁠3/4⁠
2	−⁠1/2⁠	⁠1/2⁠	⁠9/4⁠	⁠5/2⁠	⁠5/8⁠
3	−1	−⁠7/2⁠	−⁠3/4⁠	⁠15/2⁠
4	⁠5/2⁠	−⁠11/2⁠	−⁠99/4⁠
5	8	⁠77/2⁠
6	−⁠61/2⁠

From the second, the numerators of the first column are the denominators of Euler's formula. The first column is −⁠1/2⁠ ×OEIS: A163982.

The sequenceS_n has another unexpected yet important property: The denominators ofS_n+1 divide the factorialn!. In other words: the numbersT_n = S_{n + 1}n!, sometimes calledEuler zigzag numbers, are integers.

${\displaystyle T_n=1,1,1,2,5,16,61,272,1385,7936,50521,353792,\dots n=0,1,2,3,\dots }$  (OEIS: A000111). See (OEIS: A253671).

Theirexponential generating function is the sum of thesecant andtangent functions.

${\displaystyle \sum _{n=0}^{\mathrm{\infty }}{T}_n\frac{x^n}{n!}=\tan \left(\frac{\pi }{4}+\frac{x}{2}\right)=\sec x+\tan x}$ .

Thus the above representations of the Bernoulli and Euler numbers can be rewritten in terms of this sequence as