A151751 Triangle of coefficients of generalized Bernoulli polynomials associated with a Dirichlet character modulus 8.

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

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!.

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. Springer-Verlag.

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)^(n-k-1)*C(2*n,2*k)*2*(n-k)*A000464(n-k-1);

(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);

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^(n-1)*{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*(n-j).

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(n-1); 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 k-th powers twisted by the function X(n). For the present case the result is

(11)... sum{n = 0..8*N-1} X(n)*n^k = 1^k-3^k-5^k+7^k- ... +(8*N-1)^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(n-1) = (-1)^(1+binomial(n,2))*n* 2^(n-2)*A000111(n-1).

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^(n-1)*(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