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

0,3

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

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)).

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 *)

Crossrefs

Cf. A000111, A027641, A000367, A185424.

Sequence of denominators is A002445.

Sequence in context: A009762 A017952 A055433 * A009728 A027532 A021394

Adjacent sequences: A185422 A185423 A185424 * A185426 A185427 A185428

Keyword

nonn,easy

Author

Peter Bala, Feb 18 2011

Status

approved