%I
%S 2,44,2166,196888,28730410,6148123332,1813990148894,
%T 705775346640176,350112935442888018,215681051222514096220,
%U 161537815119247080938182,144555133640020128085896264,152323571317104251881943249786
%N Generalized Bernoulli numbers B_n(X,0), X a Dirichlet character modulus 8.
%C Let X be a periodic arithmetical function with period m. The generalized Bernoulli polynomials B_n(X,x) attached to X are defined by means of the generating function
%C (1)... t*exp(t*x)/(exp(m*t)1) * sum {r = 0..m1} X(r)*exp(r*t)
%C = sum {n = 0..inf} B_n(X,x)*t^n/n!.
%C The values B_n(X,0) are generalizations 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
%C (2)... X(8*n+1) = X(8*n+7) = 1; X(8*n+3) = X(8*n+5) = 1; X(2*n) = 0.
%C The oddindexed generalized Bernoulli numbers B_(2*n+1)(X,0) vanish. The current sequence lists the evenindexed values B_(2*n)(X,0).
%C The coefficients of the generalized Bernoulli polynomials B_n(X,x) are listed in A151751.
%D H. Cohen, Number Theory  Volume II: Analytic and Modern Tools, Graduate Texts in Mathematics. SpringerVerlag.
%F (1)... a(n) = (1)^(n+1)*2*n*A000464(n1).
%F The sequence of generalized Bernoulli numbers
%F (2)... [B_n(X,0)]n>=2 = [2,0,44,0,2166,0,...]
%F has the e.g.f.
%F (3)... t*(exp(t)exp(3*t)exp(5*t)+exp(7*t))/(exp(8*t)1),
%F which simplifies to
%F (4)... t*sinh(t)/cosh(2*t) = 2*t^2/2!  44*t^4/4! + ....
%F Hence
%F (5)... B_(2*n)(X,0) = (1)^(n+1)*2*n*A000464(n1) and B_(2*n+1)(X,0) = 0.
%p #A161722
%p with(gfun):
%p G(x) := x*sinh(x)/cosh(2*x):
%p coefflist := seriestolist(series(G(x),x,30)):
%p seq((2*n)!*coefflist[2*n+1],n = 1..14];
%Y Cf. A000464, A153641, A151751.
%K easy,sign
%O 2,1
%A _Peter Bala_, Jun 18 2009
%E Crossreference corrected by _Peter Bala_, Jun 22 2009
