A318141 a(n) = numerator(n!*[z^n]((cosh(x*z) + cos(x*z))*z/(1 - exp(-z)))(1)).

2, 1, 1, 0, 29, 5, 106, 0, -41, 9, 830, 0, -88051, 13, 982, 0, -487777, 17, 5911162, 0, -164321477, 21, 114840098, 0, -31762069211, 25, 8045230726, 0, -3191301739589, 29, 1157740296233330, 0, -79766429830452749, 33, 2424608499378094, 0, -3536072031131812825213

Offset

0,1

Comments

Let p(n, x) be the polynomials given implicitly in the name. Then p(n, 0)/2 = B(n, 1) where B(n, x) are the Bernoulli polynomials. In other words: p(n, 0)/2 are the Bernoulli numbers.

Links

Table of n, a(n) for n=0..36.

Formula

a(4*n + 1) = 4*n + 1 for n >= 0.

a(4*n - 1) = 0 for n >= 1.

Example

Polynomials start: 2, 1, 1/3, 0, -1/15+2*x^4, 5*x^4, 1/21+5*x^4, 0, -1/15-(14/3)*x^4+2*x^8, 9*x^8, 5/33+10*x^4+15*x^8, 0, -691/1365-33*x^4-33*x^8+2*x^12, 13*x^12, .... Evaluated at x = 1: 2, 1, 1/3, 0, 29/15, 5, 106/21, 0, -41/15, 9, 830/33, 0, -88051/1365, 13, 982/3, 0, -487777/255, 17, 5911162/399, 0, ....

Maple

gf := (cosh(x*z)+cos(x*z))*z/(1-exp(-z)): ser := series(gf, z, 70):
seq(numer(subs(x=1, n!*coeff(ser, z, n))), n=0..36);

Mathematica

m = 36;
gf = (Cosh[x*z]+Cos[x*z])*z/(1-E^-z);
Numerator[CoefficientList[(gf/.x->1)+O[z]^(m+1), z]*Range[0, m]!] (* Jean-François Alcover, Jun 04 2019 *)

Crossrefs

Cf. A318142 (denominators).

Sequence in context: A361999 A061158 A180843 * A085979 A330986 A269166

Adjacent sequences: A318138 A318139 A318140 * A318142 A318143 A318144

Keyword

sign,frac

Author

Peter Luschny, Aug 19 2018

Status

approved