A009189 Expansion of e.g.f.: exp(cos(x)*x).

1, 1, 1, -2, -11, -24, 61, 624, 1737, -7424, -88679, -242560, 2086525, 23499776, 45950997, -1002251264, -9763133167, -2151563264, 705668046769, 5583112077312, -17356978593659, -666018502836224, -3823112141007763, 39230927775531008, 788728947108214489

Offset: 0

Author

R. H. Hardin

Keywords

• sign
• easy

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..560

Crossrefs

Cf. A003727, A352252.

Programs

Mathematica

With[{nn=30},CoefficientList[Series[Exp[Cos[x]*x],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Mar 15 2018 *)

Maxima

a(n):=(sum(binomial(n,k)*(-1)^((n-k)/2)*(1+(-1)^(n-k))/(2^(k))*sum(binomial(k,i)*(k-2*i)^(n-k),i,0,floor((k-1)/2)),k,1,n-1))+1; /* Vladimir Kruchinin, Apr 21 2011 */

PARI

my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(x*cos(x)))) \\ Seiichi Manyama, Mar 26 2022

Formula

a(n) = (sum(k=1..n-1, binomial(n,k)*(-1)^((n-k)/2)*(1+(-1)^(n-k))/(2^(k))*sum(i=0..floor((k-1)/2)), binomial(k,i)*(k-2*i)^(n-k)))+1. - Vladimir Kruchinin, Apr 21 2011

a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/2)} (-1)^k * binomial(n-1,2*k) * (2*k+1) * a(n-2*k-1). - Ilya Gutkovskiy, Mar 10 2022

Extensions

Extended with signs by Olivier Gérard, Mar 15 1997

Definition clarified and prior Mathematica program replaced by Harvey P. Dale, Mar 15 2018