A190195 Numerators of a Taylor series expansion of 1/sqrt(cosh(x)) (even powers only).

1, -1, 7, -139, 5473, -51103, 34988647, -4784061619, 17782347217, -203906055033841, 4586025046220899, -234038275571853889, 9127322584507530151393, -4621897483978366951337161, 390009953658229908025520161, -1860452328661957054823447670979, 111446346975327291562408943638981, -14050053632877769956552601074149491, 1269258883676324618437848731917951368967, -1408182090109327874242950762763137949746859

Offset

0,3

Links

Table of n, a(n) for n=0..19.

Philippe Flajolet, Xavier Gourdon, and Philippe Dumas, Mellin transforms and asymptotics: harmonic sums, Special volume on mathematical analysis of algorithms. Theoret. Comput. Sci. 144 (1995), no. 1-2, 3-58.

Formula

a(n) = numerator(b(n)), where b(n) = Sum_{k=1..n} b(n-k)*(k/(2*n)-1)/(2*k)!, with b(0)=1. - Tani Akinari, Sep 17 2023

a(n) = numerator((-1)^n*A186491(n)/(4^n*(2*n)!)). - Andrew Howroyd, Sep 19 2023

Example

1/sqrt(cosh(x)) = 1 - (1/4)*x^2 + (7/96)*x^4 - (139/5760)*x^6 + (5473/645120)*x^8 - (51103/16588800)*x^10 + ...

Maple

a:= n-> numer(coeff(series(1/sqrt(cosh(x)), x, 2*n+1), x, 2*n)):
seq(a(n), n=0..19);  # Alois P. Heinz, Sep 19 2023

Prog

(Maxima) b[n]:=if n=0 then 1 else sum(b[n-k]*(k/n/2-1)/(2*k)!, k, 1, n)$ a[n]:=num(b[n])$
makelist(a[n], n, 0, 20); /* Tani Akinari, Sep 17 2023 */

Crossrefs

Cf. A190196 (denominators), A186491.

Sequence in context: A274525 A221375 A351334 * A126156 A082162 A280629

Adjacent sequences: A190192 A190193 A190194 * A190196 A190197 A190198

Keyword

sign,frac

Author

N. J. A. Sloane, May 05 2011

Status

approved