BernoulliB

The nth Bernoulli number, a rational sequence appearing in power-sum (Faulhaber) formulas and series expansions.

✓ checked against that system · ! checked, with disagreements (hover a chip)WikipediaBernoulli number↗MathWorldBernoulliNumber↗WikidataQ694114↗nLabBernoulli number↗DLMF24.2.i↗FungrimBernoulliB✓↗Fungrim entryOEISA027641↗A027642↗Wolfram LanguageBernoulliB✓↗

BernoulliB(n)the nth Bernoulli number $B_n$.

SageMathbernoulli(n)✓↗SymPybernoulli(n)↗mpmathbernoulli(n)Juliabernoulli(Int(n))

BernoulliB(n, x)the Bernoulli polynomial $B_n(x)$.

SageMathbernoulli_polynomial(x, n)SymPybernoulli(n, x)↗mpmathbernpoly(n, x)Juliaenumeratio_bernoulli_poly(n, x)

Typed signatures

Type	Package	Applies
(integer) -> rational	compute-engine	always
(integer, number?) -> number	analyticreplaces compute-engine	always

Domain: Sequences

Details

• Defined by the generating function $\dfrac{t}{e^t-1} = \sum_{n=0}^{\infty} B_n \dfrac{t^n}{n!}$.
• $B_n = 0$ for every odd $n > 1$; $B_1 = -\frac12$ under this (the "$B_n^-$") convention.
• Drive Faulhaber's formula for power sums, $\sum_{k=0}^{n-1} k^p = \frac{1}{p+1}\sum_{j=0}^{p}\binom{p+1}{j} B_j\, n^{p+1-j}$. See Binomial.
• Related to the Riemann zeta function: $\zeta(-n) = -\dfrac{B_{n+1}}{n+1}$ for $n \ge 1$, and $\zeta(2n) = \dfrac{(-1)^{n+1} B_{2n} (2\pi)^{2n}}{2\,(2n)!}$.
• compute-engine only defines $B_n$ for nonnegative integer $n$. With a second argument, BernoulliB(n, x) is the Bernoulli polynomial $B_n(x)$, as in Wolfram — the same as BernoulliPolynomial.

Plot

Examples

1 more kept as data, checked against the oracles

Basic #

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In$\mathrm{BernoulliB}(0)$Out$1$

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In$\mathrm{BernoulliB}(1)$Out$-\frac{1}{2}$

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In$\mathrm{BernoulliB}(2)$Out$\frac{1}{6}$

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In$\mathrm{BernoulliB}(4)$Out$-\frac{1}{30}$

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In$\mathrm{BernoulliB}(12)$Out$-\frac{691}{2\,730}$

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In$\mathrm{BernoulliB}(10)$Out$\frac{5}{66}$

Scope #

Exact rationals at any index

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In$\mathrm{BernoulliB}(60)$Out$-\frac{1\,215\,233\,140\,483\,755\,572\,040\,304\,994\,079\,820\,246\,041\,491}{56\,786\,730}$

Listable: $B_n$ threads over a list of indices

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In$\mathrm{BernoulliB}(\bigl\lbrack2, 4, 6\bigr\rbrack)$Out$\bigl\lbrack\frac{1}{6}, -\frac{1}{30}, \frac{1}{42}\bigr\rbrack$

The Bernoulli polynomial $B_4(x) = x^4 - 2x^3 + x^2 - \frac{1}{30}$

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In$\mathrm{BernoulliB}(4, x)$Out$x^4-2x^3+x^2-\frac{1}{30}$

The Bernoulli polynomial at a rational point: $B_2(\frac12) = -\frac{1}{12}$

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In$\mathrm{BernoulliB}(2, \frac{1}{2})$Out$-\frac{1}{12}$

With a second argument, the Bernoulli polynomial $B_n(x)$ — BernoulliPolynomial under Wolfram's name

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In$\mathrm{BernoulliB}(1, x)$Out$x-\frac{1}{2}$

Threads element-wise over a list, as Wolfram's Listable heads do

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In$\mathrm{BernoulliB}(\bigl\lbrack1, 2, 3\bigr\rbrack)$Out$\bigl\lbrack-\frac{1}{2}, \frac{1}{6}, 0\bigr\rbrack$

Applications #

Faulhaber's formula recovers $\sum_{k=0}^{4} k^2 = 30$ from $B_0, B_1, B_2$ weighted by $\binom{3}{j}$. See Binomial

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In$\frac{1}{3}(\binom{3}{0}\mathrm{BernoulliB}(0)×5^3+\binom{3}{1}\mathrm{BernoulliB}(1)×5^2+\binom{3}{2}\mathrm{BernoulliB}(2)×5^1)$Out$30$

Properties #

von Staudt–Clausen: the denominator of $B_{2n}$ is the product of the primes $p$ with $(p-1) \mid 2n$, here $2 \cdot 3 \cdot 5 \cdot 7 \cdot 13$

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In$\mathrm{Denominator}(\mathrm{BernoulliB}(12))$Out$2\,730$

$\zeta(-n) = -\frac{B_{n+1}}{n+1}$, here $\zeta(-3) = \frac{1}{120}$. See Zeta

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In$ζ(-3)=-\frac{\mathrm{BernoulliB}(4)}{4}$Out$\mathrm{True}$

The recurrence $\sum_{k=0}^{n-1} \binom{n}{k} B_k = 0$, here at $n = 5$

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In$\binom{5}{0}\mathrm{BernoulliB}(0)+\binom{5}{1}\mathrm{BernoulliB}(1)+\binom{5}{2}\mathrm{BernoulliB}(2)+\binom{5}{3}\mathrm{BernoulliB}(3)+\binom{5}{4}\mathrm{BernoulliB}(4)$Out$0$

$B_n = 0$ for every odd $n > 1$

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In$\mathrm{BernoulliB}(5)=0$Out$\mathrm{True}$

Odd-index Bernoulli numbers beyond 1 vanish

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In$\mathrm{BernoulliB}(3)$Out$0$

Known value · Wolfram MathematicalFunctionData, BernoulliB

$B_5 = 0$: odd Bernoulli numbers past $B_1$ vanish

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In$\mathrm{BernoulliB}(5)$Out$0$

Known value · Wolfram MathematicalFunctionData, BernoulliB

$B_6 = \frac{1}{42}$

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In$\mathrm{BernoulliB}(6)$Out$\frac{1}{42}$

Known value · Wolfram MathematicalFunctionData, BernoulliB

$B_7 = 0$, the odd-index zeros continue

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In$\mathrm{BernoulliB}(7)$Out$0$

Known value · Wolfram MathematicalFunctionData, BernoulliB

$B_8 = -\frac{1}{30}$

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In$\mathrm{BernoulliB}(8)$Out$-\frac{1}{30}$

Known value · Wolfram MathematicalFunctionData, BernoulliB

$B_9 = 0$, while the even ones alternate in sign

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In$\mathrm{BernoulliB}(9)$Out$0$

Known value · Wolfram MathematicalFunctionData, BernoulliB

The Bernoulli polynomial at 0 is the Bernoulli number $B_2$

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In$\mathrm{BernoulliB}(2, 0)$Out$\frac{1}{6}$

Known value · Wolfram MathematicalFunctionData, BernoulliB:Polynomial

Odd Bernoulli polynomials vanish at 0

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In$\mathrm{BernoulliB}(3, 0)$Out$0$

Known value · Wolfram MathematicalFunctionData, BernoulliB:Polynomial

$B_2(1) = B_2$: for $n \ge 2$ the polynomial agrees at 0 and 1

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In$\mathrm{BernoulliB}(2, 1)$Out$\frac{1}{6}$

Known value · Wolfram MathematicalFunctionData, BernoulliB:Polynomial

Odd Bernoulli polynomials vanish at 1

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In$\mathrm{BernoulliB}(3, 1)$Out$0$

Known value · Wolfram MathematicalFunctionData, BernoulliB:Polynomial

The index-0 polynomial is the constant 1

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In$\mathrm{BernoulliB}(0, \frac{1}{6})$Out$1$

Known value · Wolfram MathematicalFunctionData, BernoulliB:Polynomial

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In$\sum_{k=0}^2(k+3)\binom{3}{k}\mathrm{BernoulliB}(k+2)$Out$0$

Known value · Wolfram MathematicalFunctionData, BernoulliB, FunctionalEquations

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In$\sum_{k=2}^1\frac{\mathrm{BernoulliB}(k)\mathrm{BernoulliB}(3-k)}{k×(3-k)}-\sum_{k=2}^1\frac{\binom{3}{k}\mathrm{BernoulliB}(k)\mathrm{BernoulliB}(3-k)}{k×(3-k)}$Out$0$

Known value · Wolfram MathematicalFunctionData, BernoulliB, FunctionalEquations

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In$\sum_{k=0}^3(k+3)\binom{4}{k}\mathrm{BernoulliB}(k+2)$Out$-\frac{1}{2}$

Known value · Wolfram MathematicalFunctionData, BernoulliB, FunctionalEquations

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In$\sum_{k=2}^2\frac{\mathrm{BernoulliB}(k)\mathrm{BernoulliB}(4-k)}{k×(4-k)}-\sum_{k=2}^2\frac{\binom{4}{k}\mathrm{BernoulliB}(k)\mathrm{BernoulliB}(4-k)}{k×(4-k)}$Out$-\frac{5}{144}$

Known value · Wolfram MathematicalFunctionData, BernoulliB, FunctionalEquations

Possible issues #

$B_1(1) = +\frac12$ is the other convention for $B_1$

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In$\mathrm{BernoulliB}(1, 1)$Out$\frac{1}{2}$

Bernoulli numbers are only defined for $n \ge 0$; a negative index is left unevaluated

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In$\mathrm{BernoulliB}(-1)$Out$\mathrm{BernoulliB}(-1)$

Neat examples #

Numerators and denominators grow quickly: $B_{20} = -\frac{174611}{330}$

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In$\mathrm{BernoulliB}(20)$Out$-\frac{174\,611}{330}$

A full degree-12 Bernoulli polynomial in $x$

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In$\mathrm{BernoulliB}(12, x)$Out$x^{12}-6x^{11}+11x^{10}-\frac{33x^8}{2}+22x^6-\frac{33x^4}{2}+5x^2-\frac{691}{2\,730}$

About

Examples	37
Checked against	Wolfram Language 37 agreeSageMath 31 agree6 errorSymPy 30 agree5 differ1 unevaluatedmpmath 27 agree1 errormathlib4 18 agree4 errorJulia 28 agree1 error
Crosswalk	WikipediaMathWorldWikidatanLabDLMFFungrimFungrim entry ×17OEIS ×2Wolfram LanguageSageMath ×2SymPy ×2mpmath ×2Julia ×2

Implementation

mappedwolframpackages/oracle/src/mappings.ts

mappedsympypackages/oracle/src/mappings.ts

mappedsagepackages/oracle/src/mappings.ts

mappedsympypackages/oracle/src/mappings.ts

mappedmpmathpackages/oracle/src/mappings.ts

mappedmpmathpackages/oracle/src/mappings.ts

mappedsagepackages/oracle/src/mappings.ts

mappedjuliapackages/oracle/src/mappings.ts

mappedjuliapackages/oracle/src/mappings.ts

mappedmathlib4packages/oracle/src/mappings.ts

Mathlib's bernoulli (not bernoulli', which has B(1) = +1/2) takes B(1) = -1/2, as Wolfram does; natural n only.