login
Bisection of A185424. Numerators of even-indexed generalized Bernoulli numbers associated with the zigzag numbers A000111.
2

%I #16 Jul 07 2017 03:45:30

%S 1,1,19,253,3319,222557,422152729,59833795,439264083023,

%T 76632373664299,4432283799315809,317829581058418253,

%U 1297298660169509319229,696911453333335463719,28877308885785768720478751,157040990105362922778773747849

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

%C 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 The present sequence lists the numerators of ZB(2*n) for n>=0.

%H G. C. Greubel, <a href="/A185425/b185425.txt">Table of n, a(n) for n = 0..235</a>

%F (1)... 1/2*log(sec(t)+tan(t))*(1+sin(t)+cos(t))/(1+sin(t)-cos(t))

%F = Sum_{n >= 0} ZB(2*n)*t^(2*n)/(2*n)!

%F = 1 + (1/6)*t^2/2! + (19/30)*t^4/4! + (253/42)*t^6/6! + ....

%F (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 (3)... a(n) = numerator(ZB(2*n)).

%p #A185425

%p a := n - > (-1)^n*add (binomial(2*n,2*k)/(2*k+1)* bernoulli(2*n-2*k)*

%p euler(2*k), k = 0..n):

%p seq(numer(a(n)), n = 0..20);

%t 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 Cf. A000111, A027641, A000367, A185424.

%Y Sequence of denominators is A002445.

%K nonn,easy

%O 0,3

%A _Peter Bala_, Feb 18 2011