A009256 - OEIS

%I #24 Aug 08 2023 02:12:44

%S 1,2,28,872,47248,3907232,454886848,70597546112,14042505449728,

%T 3475021574246912,1045247734061145088,375054668796817221632,

%U 158085597663328138006528,77269840864693331267919872

%N Expansion of e.g.f. exp(tan(x)^2) (even powers only).

%H Robert Israel, <a href="/A009256/b009256.txt">Table of n, a(n) for n = 0..235</a>

%F a(n) = Sum_{k=1..n} (Sum_{j=2*k..2*n} binomial(j-1,2*k-1)*j!*2^(2*n-j)*(-1)^(n+k+j)*Stirling2(2*n,j)/k!). - _Vladimir Kruchinin_, Jun 06 2011

%F a(n) ~ (2*n)! * 2^(2*n+1/3) * exp(-2/3 + 4/(3*Pi^2) + (2^(4/3)*n^(1/3) + 3*n^(2/3)*(2*Pi)^(2/3))/Pi^(4/3)) / (sqrt(3) * n^(2/3) * Pi^(2*n+5/6)). - _Vaclav Kotesovec_, Jan 24 2015

%p S:= series(exp(tan(x)^2),x,31):

%p seq(coeff(S,x,j)*j!,j=0..30,2); # _Robert Israel_, Aug 07 2023

%t Exp[ Tan[ x ]^2 ] (* Even Part *)

%t nn = 20; Table[(CoefficientList[Series[E^Tan[x]^2, {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):=sum((sum(binomial(j-1,2*k-1)*j!*2^(2*n-j)*(-1)^(n+k+j)*stirling2(2*n,j),j,2*k,2*n))/k!,k,1,n); /* _Vladimir Kruchinin_, Jun 06 2011 */

%K nonn

%O 0,2

%A _R. H. Hardin_

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