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

1, 1, 19, 253, 3319, 222557, 422152729, 59833795, 439264083023, 76632373664299, 4432283799315809, 317829581058418253, 1297298660169509319229, 696911453333335463719, 28877308885785768720478751, 157040990105362922778773747849

Offset: 0

Author

Peter Bala, Feb 18 2011

Keywords

• nonn
• easy

Comments

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).

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

Links

• G. C. Greubel, Table of n, a(n) for n = 0..235

Crossrefs

Cf. A000111, A027641, A000367, A185424.

Sequence of denominators is A002445.

Programs

Maple

#A185425
a := n - > (-1)^n*add (binomial(2*n,2*k)/(2*k+1)* bernoulli(2*n-2*k)*
euler(2*k), k = 0..n):
seq(numer(a(n)), n = 0..20);

Mathematica

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 *)

Formula

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

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

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

(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).

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