A157811 Numerator of Bernoulli(n, -2/3).

1, -7, 23, -35, 973, -245, 7943, -1295, 31813, -7721, 288715, -13475, 128296423, -882557, -4891999, 33870025, 26217383381, -2149340753, -2830613025019, 167302324405, 101475278720663, -16020469382309, -4469247530896841, 1848020660952865, 11126033993150564743

Offset

0,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..250

Peter Luschny, Generalized Bernoulli numbers.

Example

From Peter Luschny, Mar 26 2021: (Start)
The rational numbers given in the definition start:
1, -7/6, 23/18, -35/27, 973/810, -245/243, 7943/10206, -1295/2187, 31813/65610, -7721/19683, 288715/1299078, -13475/177147, 128296423/483611310, ...
The generalized Bernoulli numbers defined in the Luschny link are different:
1, -7/2, 23/2, -35, 973/10, -245, 7943/14, -1295, 31813/10, -7721, 288715/22, -13475, 128296423/910, -882557, -4891999/2, ... The denominators of these numbers are in A285068. (End)

Mathematica

Table[Numerator[BernoulliB[n, -2/3]], {n, 0, 50}] (* Vincenzo Librandi, Mar 16 2014 *)

Prog

(SageMath) # Generalized Bernoulli polynomials
def gen_bernoulli_polynomial(n, m, x):
    p = sum(sum(sum(((-1)^(n-v)/(j+1))*binomial(n, k)*binomial(j, v)*(m*(v-x))^k
        for v in (0..j)) for j in (0..k)) for k in (0..n))
    return expand(p)
# Generalized Bernoulli numbers
def gen_bernoulli_number(n, m): return gen_bernoulli_polynomial(n, m, 1)
print([numerator((-1)^n*gen_bernoulli_number(n, 3)) for n in range(23)]) # Peter Luschny, Mar 26 2021

Crossrefs

For denominators see A157800.

The denominators of the generalized Bernoulli numbers are A285068.

Sequence in context: A287309 A275777 A329931 * A341284 A227421 A098029

Adjacent sequences: A157808 A157809 A157810 * A157812 A157813 A157814

Keyword

sign,frac

Author

N. J. A. Sloane, Nov 08 2009

Status

approved