A012022 - OEIS

%I #30 Feb 03 2018 16:39:48

%S 1,-5,129,-7965,903105,-163451925,43259364225,-15764670046125,

%T 7571150452490625,-4634731528895593125,3522824632122301130625,

%U -3255279003622294051528125,3593928024032353882700450625

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

%H Vincenzo Librandi, <a href="/A012022/b012022.txt">Table of n, a(n) for n = 0..100</a>

%F a(n) = (2*n+1)!*(-1)^n*Sum_{j=1..n+1} binomial((2*n-1)/2, n+1-j) /(2*j-1). - _Vladimir Kruchinin_, May 19 2011

%F E.g.f.: Sum_{n >= 0} a(n)*x^(2n+1)/(2n+1)! = arctan(sin(arctan(x))).

%F a(n) = (2*n+1)! * [x^(2*n+1)] arctan(sin(arctan(x))).

%t With[{nn=30},Take[CoefficientList[Series[ArcTan[Sin[ArcTan[x]]],{x,0,nn}],x] Range[0,nn-1]!,{2,-1,2}]] (* _Harvey P. Dale_, Dec 11 2013 *)

%o (Maxima)

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

%K sign

%O 0,2

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