A009250 - OEIS

%I #18 Jan 27 2018 06:31:50

%S 1,2,16,302,10456,564842,43545676,4528889822,610057244176,

%T 103185102761042,21388501828276756,5328050642207280902,

%U 1569616725144816645016,539516138161105990193402

%N E.g.f. exp(tan(x)*sin(x)) (even powers only).

%F a(n) = 2*Sum(k=1..2*n, Sum(t=0..n-k, binomial(2*n,2*t+k)*((Sum(j=k..2*n-2*t-k, binomial(j-1,k-1)*j!*Stirling2(2*n-2*t-k,j)*(-1)^(n+j)*2^(-2*t-k+2*n-j)))*Sum(i=0..k/2, (2*i-k)^(2*t+k)*binomial(k,i)*(-1)^(k-i))))/(2^k*k!)), n>0, a(0)=1. - _Vladimir Kruchinin_, Jun 30 2011

%F a(n) ~ (2*n)! * 2^(2*n-1/2) * exp(1/Pi + 4*sqrt(n/Pi)) / (n^(3/4) * Pi^(2*n+3/4)) * (1 - (5*Pi^2-2) / (12*Pi^(3/2)*sqrt(n))). - _Vaclav Kotesovec_, Jan 24 2015

%t nn = 20; Table[(CoefficientList[Series[E^(Sin[x]*Tan[x]), {x, 0, 2*nn}], x] * Range[0, 2*nn]!)[[n]], {n, 1, 2*nn+1, 2}] (* _Vaclav Kotesovec_, Jan 24 2015 *)

%o (Maxima)

%o a(n):=if n=0 then 1 else 2*sum(sum(binomial(2*n,2*t+k)*((sum(binomial(j-1,k-1)*j!*stirling2(2*n-2*t-k,j)*(-1)^(n+j)*2^(-2*t-k+2*n-j),j,k,2*n-2*t-k))*sum((2*i-k)^(2*t+k)*binomial(k,i)*(-1)^(k-i),i,0,k/2)),t,0,n-k)/(2^k*k!),k,1,2*n); /* _Vladimir Kruchinin_, Jun 30 2011 */

%K nonn

%O 0,2

%A _R. H. Hardin_

%E Extended and signs tested by _Olivier Gérard_, Mar 15 1997