A009119 Expansion of e.g.f. cos(x/cosh(x)) (even powers only).

1, -1, 13, -301, 11705, -698521, 59340997, -6782462597, 1000434618609, -184576848771889, 41577074746699261, -11216502744649033437, 3567416307426404300713, -1320192785381894987925961, 562163981454375064332029365, -272809563505907130928868599861

Offset: 0

Author

R. H. Hardin

Keywords

• sign

Links

• G. C. Greubel, Table of n, a(n) for n = 0..238

Programs

Mathematica

With[{nn=30},Take[CoefficientList[Series[Cos[x/Cosh[x]],{x,0,nn}],x] Range[ 0,nn]!,{1,-1,2}]] (* Harvey P. Dale, Jul 07 2017 *)

Maxima

a(n):=2*sum(binomial(2*n,2*k)*sum(binomial(k+j-1,j)*4^(n-k-j)*sum((i-j)^(2*n-2*k)*binomial(2*j,i)*(-1)^(k+j-i),i,0,j),j,0,(n-k)),k,1,n-1)+(-1)^n; /* Vladimir Kruchinin, Jun 16 2011 */

PARI

x='x+O('x^50); v=Vec(serlaplace(cos(x/cosh(x)))); vector(#v\2,n,v[2*n-1]) \\ G. C. Greubel, Jul 26 2018

Formula

a(n) = 2*Sum_{k=1..n-1} binomial(2*n,2*k)*Sum_{j=0..(n-k)} binomial(k+j-1,j)*4^(n-k-j)*Sum_{i=0..j} (i-j)^(2*n-2*k)*binomial(2*j,i)*(-1)^(k+j-i) +(-1)^n. - Vladimir Kruchinin, Jun 16 2011

Extensions

Extended with signs by Olivier Gérard, Mar 15 1997