

A151751


Triangle of coefficients of generalized Bernoulli polynomials associated with a Dirichlet character modulus 8.


3



2, 0, 6, 44, 0, 12, 0, 220, 0, 20, 2166, 0, 660, 0, 30, 0, 15162, 0, 1540, 0, 42, 196888, 0, 60648, 0, 3080, 0, 56, 0, 1771992, 0, 181944, 0, 5544, 0, 72, 28730410, 0, 8859960, 0, 454860, 0, 9240, 0, 90
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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..m1} X(r)*exp(r*t)
= sum {n = 0..inf} B_n(X,x)*t^n/n!.
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
(2)... X(8*n+1) = X(8*n+7) = 1; X(8*n+3) = X(8*n+5) = 1; X(2*n) = 0.
Cf. A153641.


REFERENCES

H. Cohen, Number Theory  Volume II: Analytic and Modern Tools, Graduate Texts in Mathematics. SpringerVerlag.


LINKS

Table of n, a(n) for n=2..46.


FORMULA

TABLE ENTRIES
(1)... T(2*n,2*k+1) = 0, T(2*n+1,2*k) = 0;
(2)... T(2*n,2*k) = (1)^(nk1)*C(2*n,2*k)*2*(nk)*A000464(nk1);
(3)... T(2*n+1,2*k+1) = (1)^(nk1)*C(2*n+1,2*k+1)*2*(nk)*A000464(nk1);
where C(n,k) = binomial(n,k).
GENERATING FUNCTION
The e.g.f. for these generalized Bernoulli polynomials is
(4)... t*exp(x*t)*(exp(t)exp(3*t)exp(5*t)+exp(7*t))/(exp(8*t)1)
= 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! + ....
In terms of the ordinary Bernoulli polynomials B_n(x)
(5)... B_n(X,x) = 8^(n1)*{B_n((x+1)/8)  B_n((x+3)/8)  B_n((x+5)/8) + B_n((x+7)/8)}.
The B_n(X,x) are Appell polynomials of the form
(6)... B_n(X,x) = sum {j = 0..n} binomial(n,j)*B_j(X,0)*x*(nj).
The sequence of generalized Bernoulli numbers
(7)... [B_n(X,0)]n>=2 = [2,0,44,0,2166,0,...]
has the e.g.f.
(8)... t*(exp(t)exp(3*t)exp(5*t)+exp(7*t))/(exp(8*t)1),
which simplifies to
(9)... t*sinh(t)/cosh(2*t).
Hence
(10)... B_(2*n)(X,0) = (1)^(n+1)*2*n*A000464(n1); B_(2*n+1)(X,0) = 0.
The sequence {B_(2*n)(X,0)}n>=2 is A161722.
RELATION WITH TWISTED SUMS OF POWERS
The generalized Bernoulli polynomials may be used to evaluate sums of kth powers twisted by the function X(n). For the present case the result is
(11)... sum{n = 0..8*N1} X(n)*n^k = 1^k3^k5^k+7^k ... +(8*N1)^k
= [B_(k+1)(X,8*N)  B_(k+1)(X,0)]/(k+1)
For the proof, apply [Cohen, Corollary 9.4.17 with m = 8 and x = 0].
MISCELLANEOUS
(12)... Row sums [2, 6, 32, ...] = (1)^(1+binomial(n,2))*A109572(n)
= (1)^(1+binomial(n,2))*n*A000828(n1) = (1)^(1+binomial(n,2))*n* 2^(n2)*A000111(n1).


EXAMPLE

The triangle begins
n\k........0.......1........2.......3......4.......5.......6
=============================================================
.2.........2
.3.........0.......6
.4.......44.......0.......12
.5.........0....220........0......20
.6......2166.......0.....660.......0......30
.7.........0...15162........0...1540.......0.....42
.8...196888.......0....60648.......0...3080......0......56
...


MAPLE

with(gfun):
for n from 2 to 10 do
Genbernoulli(n, x) := 8^(n1)*(bernoulli(n, (x+1)/8)bernoulli(n, (x+3)/8)bernoulli(n, (x+5)/8)+bernoulli(n, (x+7)/8));
seriestolist(series(Genbernoulli(n, x), x, 10))
end do;


CROSSREFS

Cf. A000111, A000464, A000828, A001586, A109572, A153641, A161722.
Sequence in context: A278746 A280217 A079203 * A196354 A305620 A294470
Adjacent sequences: A151748 A151749 A151750 * A151752 A151753 A151754


KEYWORD

easy,tabl,sign


AUTHOR

Peter Bala, Jun 17 2009


STATUS

approved



