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A185425 Bisection of A185424. Numerators of even-indexed generalized Bernoulli numbers associated with the zigzag numbers A000111. 2
1, 1, 19, 253, 3319, 222557, 422152729, 59833795, 439264083023, 76632373664299, 4432283799315809, 317829581058418253, 1297298660169509319229, 696911453333335463719, 28877308885785768720478751, 157040990105362922778773747849 (list; graph; refs; listen; history; text; internal format)
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

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Last modified May 17 12:01 EDT 2021. Contains 343971 sequences. (Running on oeis4.)