2,1

Let X be a periodic arithmetical function with period m. The generalised Bernoulli polynomials B_n(X,x) attached to X are defined by means of the generating function

(1)... t*exp(t*x)/(exp(m*t)-1) * sum {r = 0..m-1} X(r)*exp(r*t)

= sum {n = 0..inf} B_n(X,x)*t^n/n!.

The values B_n(X,0) are generalisations of the Bernoulli numbers (case X = 1). For the theory and properties of these polynomials and numbers see [Cohen, Section 9.4]. In the present case, X is chosen to be the Dirichlet character modulus 8 given by

(2)... X(8*n+1) = X(8*n+7) = 1; X(8*n+3) = X(8*n+5) = -1; X(2*n) = 0.

The odd indexed generalised Bernoulli numbers B_(2*n+1)(X,0) vanish. The current sequence lists the even indexed values B_(2*n)(X,0).

The coefficients of the generalised Bernoulli polynomials B_n(X,x) are listed in A151751.

H. Cohen, Number Theory - Volume II: Analytic and Modern Tools, Graduate Texts in Mathematics. Springer-Verlag.

(1)... a(n) = (-1)^(n+1)*2*n*A000464(n-1).

The sequence of generalised Bernoulli numbers

(2)... [B_n(X,0)]n>=2 = [2,0,-44,0,2166,0,...]

has the e.g.f.

(3)... t*(exp(t)-exp(3*t)-exp(5*t)+exp(7*t))/(exp(8*t)-1),

which simplifies to

(4)... t*sinh(t)/cosh(2*t) = 2*t^2/2! - 44*t^4/4! + ....

Hence

(5)... B_(2*n)(X,0) = (-1)^(n+1)*2*n*A000464(n-1) and B_(2*n+1)(X,0) = 0.

#A161722

with(gfun):

G(x) := x*sinh(x)/cosh(2*x):

coefflist := seriestolist(series(G(x), x, 30)):

seq((2*n)!*coefflist[2*n+1], n = 1..14];

A000464, A153641, A151751.

Sequence in context: A140795 A161744 A054732 this_sequence A054914 A161745 A048566

Adjacent sequences: A161719 A161720 A161721 this_sequence A161723 A161724 A161725

easy,sign

Peter Bala (pbala(AT)talktalk.net), Jun 18 2009

Cross-reference corrected by Peter Bala (pbala(AT)talktalk.net), Jun 22 2009