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%I #4 Jul 11 2012 14:45:14
%S 2,0,6,-44,0,12,0,-220,0,20,2166,0,-660,0,30,0,15162,0,-1540,0,42,
%T -196888,0,60648,0,-3080,0,56,0,-1771992,0,181944,0,-5544,0,72,
%U 28730410,0,-8859960,0,454860,0,-9240,0,90
%N Triangle of coefficients of generalized Bernoulli polynomials associated with 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 For the theory and properties of these polynomials 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 Cf. A153641.
%D H. Cohen, Number Theory - Volume II: Analytic and Modern Tools, Graduate Texts in Mathematics. Springer-Verlag.
%F TABLE ENTRIES
%F (1)... T(2*n,2*k+1) = 0, T(2*n+1,2*k) = 0;
%F (2)... T(2*n,2*k) = (-1)^(n-k-1)*C(2*n,2*k)*2*(n-k)*A000464(n-k-1);
%F (3)... T(2*n+1,2*k+1) = (-1)^(n-k-1)*C(2*n+1,2*k+1)*2*(n-k)*A000464(n-k-1);
%F where C(n,k) = binomial(n,k).
%F GENERATING FUNCTION
%F The e.g.f. for these generalized Bernoulli polynomials is
%F (4)... t*exp(x*t)*(exp(t)-exp(3*t)-exp(5*t)+exp(7*t))/(exp(8*t)-1)
%F = sum {n = 2..inf} B_n(X,x)*t^n/n! = 2*t^2/2! + 6*x*t^3/3! + (12*x^2 - 44)*t^4/4! + ....
%F In terms of the ordinary Bernoulli polynomials B_n(x)
%F (5)... B_n(X,x) = 8^(n-1)*{B_n((x+1)/8) - B_n((x+3)/8) - B_n((x+5)/8) + B_n((x+7)/8)}.
%F The B_n(X,x) are Appell polynomials of the form
%F (6)... B_n(X,x) = sum {j = 0..n} binomial(n,j)*B_j(X,0)*x*(n-j).
%F The sequence of generalized Bernoulli numbers
%F (7)... [B_n(X,0)]n>=2 = [2,0,-44,0,2166,0,...]
%F has the e.g.f.
%F (8)... t*(exp(t)-exp(3*t)-exp(5*t)+exp(7*t))/(exp(8*t)-1),
%F which simplifies to
%F (9)... t*sinh(t)/cosh(2*t).
%F Hence
%F (10)... B_(2*n)(X,0) = (-1)^(n+1)*2*n*A000464(n-1); B_(2*n+1)(X,0) = 0.
%F The sequence {B_(2*n)(X,0)}n>=2 is A161722.
%F RELATION WITH TWISTED SUMS OF POWERS
%F The generalized Bernoulli polynomials may be used to evaluate sums of k-th powers twisted by the function X(n). For the present case the result is
%F (11)... sum{n = 0..8*N-1} X(n)*n^k = 1^k-3^k-5^k+7^k- ... +(8*N-1)^k
%F = [B_(k+1)(X,8*N) - B_(k+1)(X,0)]/(k+1)
%F For the proof, apply [Cohen, Corollary 9.4.17 with m = 8 and x = 0].
%F MISCELLANEOUS
%F (12)... Row sums [2, 6, -32, ...] = (-1)^(1+binomial(n,2))*A109572(n)
%F = (-1)^(1+binomial(n,2))*n*A000828(n-1) = (-1)^(1+binomial(n,2))*n* 2^(n-2)*A000111(n-1).
%e The triangle begins
%e n\k|........0.......1........2.......3......4.......5.......6
%e =============================================================
%e .2.|........2
%e .3.|........0.......6
%e .4.|......-44.......0.......12
%e .5.|........0....-220........0......20
%e .6.|.....2166.......0.....-660.......0......30
%e .7.|........0...15162........0...-1540.......0.....42
%e .8.|..-196888.......0....60648.......0...-3080......0......56
%e ...
%p with(gfun):
%p for n from 2 to 10 do
%p Genbernoulli(n,x) := 8^(n-1)*(bernoulli(n,(x+1)/8)-bernoulli(n,(x+3)/8)-bernoulli(n,(x+5)/8)+bernoulli(n,(x+7)/8));
%p seriestolist(series(Genbernoulli(n,x),x,10))
%p end do;
%Y Cf. A000111, A000464, A000828, A001586, A109572, A153641, A161722.
%K easy,tabl,sign
%O 2,1
%A _Peter Bala_, Jun 17 2009
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