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

%I #12 Mar 26 2021 12:17:06

%S 1,-7,23,-35,973,-245,7943,-1295,31813,-7721,288715,-13475,128296423,

%T -882557,-4891999,33870025,26217383381,-2149340753,-2830613025019,

%U 167302324405,101475278720663,-16020469382309,-4469247530896841,1848020660952865,11126033993150564743

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

%H Vincenzo Librandi, <a href="/A157811/b157811.txt">Table of n, a(n) for n = 0..250</a>

%H Peter Luschny, <a href="http://www.luschny.de/math/euler/GeneralizedBernoulliNumbers.html">Generalized Bernoulli numbers</a>.

%e From _Peter Luschny_, Mar 26 2021: (Start)

%e The rational numbers given in the definition start:

%e 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, ...

%e The generalized Bernoulli numbers defined in the Luschny link are different:

%e 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)

%t Table[Numerator[BernoulliB[n, -2/3]], {n, 0, 50}] (* _Vincenzo Librandi_, Mar 16 2014 *)

%o (SageMath) # Generalized Bernoulli polynomials

%o def gen_bernoulli_polynomial(n, m, x):

%o p = sum(sum(sum(((-1)^(n-v)/(j+1))*binomial(n,k)*binomial(j,v)*(m*(v-x))^k

%o for v in (0..j)) for j in (0..k)) for k in (0..n))

%o return expand(p)

%o # Generalized Bernoulli numbers

%o def gen_bernoulli_number(n, m): return gen_bernoulli_polynomial(n, m, 1)

%o print([numerator((-1)^n*gen_bernoulli_number(n, 3)) for n in range(23)]) # _Peter Luschny_, Mar 26 2021

%Y For denominators see A157800.

%Y The denominators of the generalized Bernoulli numbers are A285068.

%K sign,frac

%O 0,2

%A _N. J. A. Sloane_, Nov 08 2009