A012021 - OEIS

%I #18 Jan 29 2018 02:56:35

%S 1,-1,1,167,-14303,1383887,-170123615,26560717367,-5162935778879,

%T 1219537456849055,-340794504958201919,109077799391707298759,

%U -38112045733323708444959,13139859774638771226676847

%N Expansion of e.g.f.: tan(sin(arctan(x))) (odd powers only).

%H Vincenzo Librandi, <a href="/A012021/b012021.txt">Table of n, a(n) for n = 0..200</a>

%F a(n) = (2*n-1)!*sum(m=1..n, (sum(j=1..2*m-1, j!*2^(2*m-j-1)*(-1)^(n+j)*Stirling2(2*m-1,j)))*binomial((2*n-3)/2,(n-m))/(2*m-1)!), n>0. - _Vladimir Kruchinin_, May 19 2011

%e tan(sin(arctan(x))) = x - (1/3!)*x^3 + (1/5!)*x^5 + (167/7!)*x^7 - (14303/9!)*x^9 + ...

%t nn = 20; Table[(CoefficientList[Series[Tan[x/Sqrt[1 + x^2]], {x, 0, 2*nn+1}], x] * Range[0, 2*nn+1]!)[[n]], {n, 2, 2*nn, 2}] (* _Vaclav Kotesovec_, Feb 03 2015 *)

%o (Maxima)

%o a(n):=(2*n-1)!*sum((sum(j!*2^(2*m-j-1)*(-1)^(n+j)*stirling2(2*m-1,j),j,1,2*m-1))*binomial((2*n-3)/2,(n-m))/(2*m-1)!,m,1,n); /* _Vladimir Kruchinin_, May 19 2011 */

%K sign

%O 0,4

%A Patrick Demichel (patrick.demichel(AT)hp.com)