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

%I A157811 #12 Mar 26 2021 12:17:06
%S A157811 1,-7,23,-35,973,-245,7943,-1295,31813,-7721,288715,-13475,128296423,
%T A157811 -882557,-4891999,33870025,26217383381,-2149340753,-2830613025019,
%U A157811 167302324405,101475278720663,-16020469382309,-4469247530896841,1848020660952865,11126033993150564743
%N A157811 Numerator of Bernoulli(n, -2/3).
%H A157811 Vincenzo Librandi, <a href="/A157811/b157811.txt">Table of n, a(n) for n = 0..250</a>
%H A157811 Peter Luschny, <a href="http://www.luschny.de/math/euler/GeneralizedBernoulliNumbers.html">Generalized Bernoulli numbers</a>.
%e A157811 From _Peter Luschny_, Mar 26 2021: (Start)
%e A157811 The rational numbers given in the definition start:
%e A157811 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 A157811 The generalized Bernoulli numbers defined in the Luschny link are different:
%e A157811 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 A157811 Table[Numerator[BernoulliB[n, -2/3]], {n, 0, 50}] (* _Vincenzo Librandi_, Mar 16 2014 *)
%o A157811 (SageMath) # Generalized Bernoulli polynomials
%o A157811 def gen_bernoulli_polynomial(n, m, x):
%o A157811     p = sum(sum(sum(((-1)^(n-v)/(j+1))*binomial(n,k)*binomial(j,v)*(m*(v-x))^k
%o A157811         for v in (0..j)) for j in (0..k)) for k in (0..n))
%o A157811     return expand(p)
%o A157811 # Generalized Bernoulli numbers
%o A157811 def gen_bernoulli_number(n, m): return gen_bernoulli_polynomial(n, m, 1)
%o A157811 print([numerator((-1)^n*gen_bernoulli_number(n, 3)) for n in range(23)]) # _Peter Luschny_, Mar 26 2021
%Y A157811 For denominators see A157800.
%Y A157811 The denominators of the generalized Bernoulli numbers are A285068.
%K A157811 sign,frac
%O A157811 0,2
%A A157811 _N. J. A. Sloane_, Nov 08 2009