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A161722 Generalized Bernoulli numbers B_n(X,0), X a Dirichlet character modulus 8. 3


%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..m-1} 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 odd-indexed generalized Bernoulli numbers B_(2*n+1)(X,0) vanish. The current sequence lists the even-indexed 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. Springer-Verlag.

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

%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(n-1) 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 Cross-reference corrected by _Peter Bala_, Jun 22 2009

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Last modified August 8 19:29 EDT 2020. Contains 336298 sequences. (Running on oeis4.)