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

0, 2, -44, 2166, -196888, 28730410, -6148123332, 1813990148894, -705775346640176, 350112935442888018, -215681051222514096220, 161537815119247080938182, -144555133640020128085896264, 152323571317104251881943249786

Offset

0,2

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} 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

Table of n, a(n) for n=0..13.

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.

a(n) = (-1/2)*16^n*n*euler(2*n-1, 1/4) for n >= 1 after a formula of Peter Bala in A000464. - Peter Luschny, Nov 26 2020

Maple

G := x*sinh(x)/cosh(2*x): ser := series(G, x, 30):
seq((2*n)!*coeff(ser, x, 2*n), n = 0..14); # Peter Luschny, Nov 26 2020
# After an observation of F. Chapoton in A117442:
A161722 := proc(n) 4^n*add(binomial(2*n, k)*euler(k)*((x+1)/2)^(2*n-k), k=0..2*n);
coeff(%, x, 1) end: seq(A161722(n), n=0..13); # Peter Luschny, Nov 26 2020

Mathematica

terms = 13;
(CoefficientList[x(Sinh[x]/Cosh[2x]) + O[x]^(2terms+3), x] Range[0, 2terms+2]!)[[ ;; ;; 2]] (* Jean-François Alcover, Nov 16 2020 *)

Crossrefs

Cf. A000464, A153641, A151751, A117442.

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

Offset set to 0 and a(0) = 0 prepended by Peter Luschny, Nov 26 2020

Status

approved