A009210 Expansion of e.g.f.: exp(sin(x)*cos(x)).

1, 1, 1, -3, -15, -23, 177, 1253, 1057, -37103, -245471, 371085, 15691665, 76436089, -608056239, -10302629131, -20287425215, 856245051169, 8821231566145, -29959421725155, -1376333505095631, -7591883371988471, 139148719952772849

Offset: 0

Author

R. H. Hardin

Keywords

• sign
• easy

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..125

Programs

Mathematica

With[{nn=30},CoefficientList[Series[Exp[Sin[x]*Cos[x]],{x,0,nn}],x] Range[ 0,nn]!] (* Harvey P. Dale, Aug 10 2021 *)

Maxima

a(n):=sum((2^(4*j-n+1)*sum((2*i+2*j-n)^n*binomial(n-2*j,i)*(-1)^(n-j-i),i,0,((n-2*j)/2)))/(n-2*j)!,j,0,((n-1)/2)); /* Vladimir Kruchinin, May 29 2011 */

PARI

x='x+O('x^66); Vec(serlaplace(exp(sin(x)*cos(x)))) /* Joerg Arndt, May 29 2011 */

Formula

a(n) = Sum_{j=0..(n-1)/2} 2^(4*j-n+1)*(Sum_{i=0..(n-2*j)/2} (2*i+2*j-n)^n*binomial(n-2*j,i)*(-1)^(n-j-i))/(n-2*j)!, n>0, a(0)=1. - Vladimir Kruchinin, May 29 2011

a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/2)} binomial(n-1,2*k) * (-4)^k * a(n-2*k-1). - Ilya Gutkovskiy, Feb 24 2022

Extensions

Extended with signs by Olivier Gérard, Mar 15 1997

Definition corrected by Joerg Arndt, May 29 2011

Definition clarified and prior Mathematica program replaced by Harvey P. Dale, Aug 10 2021