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

%I A161722 #28 Nov 26 2020 09:38:31
%S A161722 0,2,-44,2166,-196888,28730410,-6148123332,1813990148894,
%T A161722 -705775346640176,350112935442888018,-215681051222514096220,
%U A161722 161537815119247080938182,-144555133640020128085896264,152323571317104251881943249786
%N A161722 Generalized Bernoulli numbers B_n(X,0), X a Dirichlet character modulus 8.
%C A161722 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 A161722 (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!.
%C A161722 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 A161722 (2)... X(8*n+1) = X(8*n+7) = 1; X(8*n+3) = X(8*n+5) = -1; X(2*n) = 0.
%C A161722 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 A161722 The coefficients of the generalized Bernoulli polynomials B_n(X,x) are listed in A151751.
%D A161722 H. Cohen, Number Theory - Volume II: Analytic and Modern Tools, Graduate Texts in Mathematics. Springer-Verlag.
%F A161722 (1)... a(n) = (-1)^(n+1)*2*n*A000464(n-1).
%F A161722 The sequence of generalized Bernoulli numbers
%F A161722 (2)... [B_n(X,0)]n>=2 = [2,0,-44,0,2166,0,...]
%F A161722 has the e.g.f.
%F A161722 (3)... t*(exp(t)-exp(3*t)-exp(5*t)+exp(7*t))/(exp(8*t)-1),
%F A161722 which simplifies to
%F A161722 (4)... t*sinh(t)/cosh(2*t) = 2*t^2/2! - 44*t^4/4! + ....
%F A161722 Hence
%F A161722 (5)... B_(2*n)(X,0) = (-1)^(n+1)*2*n*A000464(n-1) and B_(2*n+1)(X,0) = 0.
%F A161722 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
%p A161722 G := x*sinh(x)/cosh(2*x): ser := series(G, x, 30):
%p A161722 seq((2*n)!*coeff(ser, x, 2*n), n = 0..14); # _Peter Luschny_, Nov 26 2020
%p A161722 # After an observation of _F. Chapoton_ in A117442:
%p A161722 A161722 := proc(n) 4^n*add(binomial(2*n, k)*euler(k)*((x+1)/2)^(2*n-k), k=0..2*n);
%p A161722 coeff(%, x, 1) end: seq(A161722(n), n=0..13); # _Peter Luschny_, Nov 26 2020
%t A161722 terms = 13;
%t A161722 (CoefficientList[x(Sinh[x]/Cosh[2x]) + O[x]^(2terms+3), x] Range[0, 2terms+2]!)[[ ;; ;; 2]] (* _Jean-François Alcover_, Nov 16 2020 *)
%Y A161722 Cf. A000464, A153641, A151751, A117442.
%K A161722 easy,sign
%O A161722 0,2
%A A161722 _Peter Bala_, Jun 18 2009
%E A161722 Cross-reference corrected by _Peter Bala_, Jun 22 2009
%E A161722 Offset set to 0 and a(0) = 0 prepended by _Peter Luschny_, Nov 26 2020