cp's OEIS Frontend

A185425 Bisection of A185424. Numerators of even-indexed generalized Bernoulli numbers associated with the zigzag numbers A000111.

%I A185425 #16 Jul 07 2017 03:45:30
%S A185425 1,1,19,253,3319,222557,422152729,59833795,439264083023,
%T A185425 76632373664299,4432283799315809,317829581058418253,
%U A185425 1297298660169509319229,696911453333335463719,28877308885785768720478751,157040990105362922778773747849
%N A185425 Bisection of A185424. Numerators of even-indexed generalized Bernoulli numbers associated with the zigzag numbers A000111.
%C A185425 Let E(t) = sec(t)+tan(t) denote the generating function for the zigzag numbers A000111. The zigzag Bernoulli numbers, denoted ZB(n), are defined by means of the generating function log E(t)/(E(t)-1) = Sum_{n>=0} ZB(n)*t^n/n!. See formula (1).
%C A185425 The present sequence lists the numerators of ZB(2*n) for n>=0.
%H A185425 G. C. Greubel, <a href="/A185425/b185425.txt">Table of n, a(n) for n = 0..235</a>
%F A185425 (1)... 1/2*log(sec(t)+tan(t))*(1+sin(t)+cos(t))/(1+sin(t)-cos(t))
%F A185425 = Sum_{n >= 0} ZB(2*n)*t^(2*n)/(2*n)!
%F A185425 = 1 + (1/6)*t^2/2! + (19/30)*t^4/4! + (253/42)*t^6/6! + ....
%F A185425 (2)... ZB(2*n) = (-1)^n*Sum_{k = 0..n} binomial(2*n,2*k)/(2*k+1)* Bernoulli(2*n-2*k)*Euler(2*k).
%F A185425 (3)... a(n) = numerator(ZB(2*n)).
%p A185425 #A185425
%p A185425 a := n - > (-1)^n*add (binomial(2*n,2*k)/(2*k+1)* bernoulli(2*n-2*k)*
%p A185425 euler(2*k), k = 0..n):
%p A185425 seq(numer(a(n)), n = 0..20);
%t A185425 Numerator[Table[(-1)^n*Sum[Binomial[2*n, 2*k]*BernoulliB[2*(n - k)]* EulerE[2*k]/(2*k + 1), {k, 0, n}], {n, 0, 50}]] (* _G. C. Greubel_, Jul 06 2017 *)
%Y A185425 Cf. A000111, A027641, A000367, A185424.
%Y A185425 Sequence of denominators is A002445.
%K A185425 nonn,easy
%O A185425 0,3
%A A185425 _Peter Bala_, Feb 18 2011