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ID: INPUT{id.yaml}

Title: Bernoulli numbers

Definition: >
  The Bernoulli numbers $B_n$ are the coefficients of
  the exponential generating functon
  $\frac{t}{e^t-1} = \sum_{n=0}^\infty \frac{B_nt^n}{n!}$.

Parameters:
  n:
    type: Z
    constraints: $n \geq 0$

Comments: 
  comment-sign-convention: >
    Some authors prefer to define
    $B^+_n = (-1)^n B_n$
    as Bernoulli numbers.

Formulas: 
  formula-cosh: >
    $\sum_{n=0}^\infty \frac{B_nt^n}{n!} = 
    \frac{t}{2}\left(\coth \frac{t}{2}-1\right)$.
  formula-zeta: >
    $B_n = (-1)^{n+1} n \zeta(1-n)$ for $n\geq 1$, 
    where $\zeta$ is the Riemann zeta function.

Programs: 
  program-sage:
    language: Sage
    code: |
      numbers = [bernoulli(n) for n in [1..10]]

References:

Links:
  Wiki:
    title: "Wikipedia: Bernoulli number"
    url: https://en.wikipedia.org/wiki/Bernoulli_number

Similar tables:
  
Keywords:
  
Tags:
- sequence
- generating function
- special values

Data properties:
  type: Q
  both signs: True
  
Display properties:
  number-header: $B_n$

Numbers: INPUT{numbers.yaml}

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Bernoulli numbers

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Numbers

$n$

$B_n$

INPUT{numbers.yaml} (not shown in preview)

Definition

The Bernoulli numbers $B_n$ are the coefficients of the exponential generating functon $\frac{t}{e^t-1} = \sum_{n=0}^\infty \frac{B_nt^n}{n!}$.

Parameters

$n$

— integer ($n \geq 0$)

Formulas

(1)

$\sum_{n=0}^\infty \frac{B_nt^n}{n!} = \frac{t}{2}\left(\coth \frac{t}{2}-1\right)$.

(2)

$B_n = (-1)^{n+1} n \zeta(1-n)$ for $n\geq 1$, where $\zeta$ is the Riemann zeta function.

Comments

(3)

Some authors prefer to define $B^+_n = (-1)^n B_n$ as Bernoulli numbers.

Programs

(P1)

Sage

numbers = [bernoulli(n) for n in [1..10]]

Links

[1]

Wikipedia: Bernoulli number

Data properties

Entries are of type: rational number

both signs: True (Unknown key)