A009563 Expansion of e.g.f. sin(x/cosh(x)) (odd powers only).

1, -4, 56, -1688, 84160, -6141312, 613282944, -80158806016, 13267800137728, -2710082835353600, 669033814167273472, -196220826200422416384, 67398310755666413387776, -26784943833122921085534208, 12188704989839014070713057280, -6294241591527906725538628370432

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[Sin[x/Cosh[x]],{x,0,nn}],x] Range[ 0,nn-1]!,{2,-1,2}]] (* Harvey P. Dale, Dec 24 2017 *)

Maxima

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

PARI

my(x='x+O('x^50)); v=Vec(serlaplace(sin(x/cosh(x)))); vector((#v-1)\2 ,n,v[2*n-1]) \\ G. C. Greubel, Jan 21 2018

Formula

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

Extensions

Extended with signs by Olivier Gérard, Mar 15 1997