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 A161722 Generalized Bernoulli numbers B_n(X,0), X a Dirichlet character modulus 8. 3
 2, -44, 2166, -196888, 28730410, -6148123332, 1813990148894, -705775346640176, 350112935442888018, -215681051222514096220, 161537815119247080938182, -144555133640020128085896264, 152323571317104251881943249786 (list; graph; refs; listen; history; text; internal format)
 OFFSET 2,1 COMMENTS 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 (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 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 (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 generalized 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 generalized Bernoulli polynomials B_n(X,x) are listed in A151751. REFERENCES H. Cohen, Number Theory - Volume II: Analytic and Modern Tools, Graduate Texts in Mathematics. Springer-Verlag. LINKS FORMULA (1)... a(n) = (-1)^(n+1)*2*n*A000464(n-1). The sequence of generalized 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. MAPLE 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]; CROSSREFS Cf. A000464, A153641, A151751. Sequence in context: A208045 A267070 A054732 * A290879 A054914 A329021 Adjacent sequences:  A161719 A161720 A161721 * A161723 A161724 A161725 KEYWORD easy,sign AUTHOR Peter Bala, Jun 18 2009 EXTENSIONS Cross-reference corrected by Peter Bala, Jun 22 2009 STATUS approved

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Last modified July 7 14:34 EDT 2020. Contains 335495 sequences. (Running on oeis4.)