A009238 Expansion of e.g.f. exp(tan(sin(x))).

1, 1, 1, 2, 5, 8, 13, -64, -855, -5632, -38791, -205184, -747539, -240640, 59637061, 859820032, 9421489105, 90170851328, 573991066225, 1502445600768, -49290541346219, -1320541298393088, -20481513828195331, -272882319216148480

Offset: 0

Author

R. H. Hardin

Keywords

• sign
• easy

Programs

Mathematica

With[{nn=30},CoefficientList[Series[Exp[Tan[Sin[x]]],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Nov 25 2011 *)

Maxima

a(n):=sum(sum((((-1)^(k-m)+1)*(sum(binomial(j-1,m-1)*j!*2^(k-j-1) *stirling2(k,j)*(-1)^((m+k)/2+j),j,m,k))*((-1)^(n-k)+1)*sum((2*i-k)^n *binomial(k,i)*(-1)^((n+k)/2-i),i,0,k/2))/(2^k*k!),k,m,n)/m!,m,1,n); /* Vladimir Kruchinin, May 05 2011 */

PARI

x='x+O('x^66); /* that many terms */
egf=exp(tan(sin(x))); /* = 1 + x + 1/2*x^2 + 1/3*x^3 + 5/24*x^4 + ... */
Vec(serlaplace(egf)) /* show terms */ /* Joerg Arndt, May 05 2011 */

Formula

a(n) = Sum(m=1..n, Sum(k=m..n, (((-1)^(k-m)+1)*(Sum(j=m..k, C(j-1,m-1)*j! *2^(k-j-1) *Stirling2(k,j)*(-1)^((m+k)/2+j)))*((-1)^(n-k)+1)*Sum(i=0..k/2, (2*i-k)^n *C(k,i)*(-1)^((n+k)/2-i)))/(2^k*k!))/m!). - Vladimir Kruchinin, May 05 2011

Extensions

Extended with signs by Olivier Gérard, Mar 15 1997

Definition corrected by Joerg Arndt, May 05 2011