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
Examples
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 + ...
Links
- 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.
Programs
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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
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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 */
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