A009189 - OEIS

%I #33 Mar 26 2022 13:41:04

%S 1,1,1,-2,-11,-24,61,624,1737,-7424,-88679,-242560,2086525,23499776,

%T 45950997,-1002251264,-9763133167,-2151563264,705668046769,

%U 5583112077312,-17356978593659,-666018502836224,-3823112141007763,39230927775531008,788728947108214489

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

%H Seiichi Manyama, <a href="/A009189/b009189.txt">Table of n, a(n) for n = 0..560</a>

%F 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

%F 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

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

%o (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 */

%o (PARI) my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(x*cos(x)))) \\ _Seiichi Manyama_, Mar 26 2022

%Y Cf. A003727, A352252.

%K sign,easy

%O 0,4

%A _R. H. Hardin_

%E Extended with signs by _Olivier Gérard_, Mar 15 1997

%E Definition clarified and prior Mathematica program replaced by _Harvey P. Dale_, Mar 15 2018