A009446 - OEIS

%I #29 Jan 18 2025 09:09:36

%S 1,-4,36,-848,23824,-871872,44942912,-2860930816,213856723200,

%T -19099352929280,2010469524579328,-242177953175506944,

%U 32998102512641970176,-5069553933584348397568,870871673923373487243264,-165613871131289362192596992,34644281516790333651724337152

%N Expansion of e.g.f. sin(x*cos(x)) (odd powers only).

%H Robert Israel, <a href="/A009446/b009446.txt">Table of n, a(n) for n = 0..300</a>

%F a(n) = (-1)^n * Sum_{k=0..n} (binomial(2*n+1,2*k+1) * (Sum_{i=0..k} (2*k+1-2*i)^(2*n-2*k)*binomial(2*k+1,i)) / 2^(2*k)). - _Vladimir Kruchinin_, Jun 18 2011

%p S:= series(sin(x*cos(x)),x,60):

%p seq(coeff(S,x,i)*i!,i=1..59,2); # _Robert Israel_, Oct 09 2024

%t With[{nn=30},Take[CoefficientList[Series[Sin[x Cos[x]],{x,0,nn}],x] Range[0,nn-1]!,{2,-1,2}]] (* _Harvey P. Dale_, Oct 04 2017 *)

%o (Maxima) a(n):=(-1)^(n)*sum(binomial(2*n+1,2*k+1)*((sum((2*k+1-2*i)^(2*n-2*k)*binomial(2*k+1,i),i,0,((k)))))/(2^(2*k)),k,0,n); /* _Vladimir Kruchinin_, Jun 18 2011 */

%K sign

%O 0,2

%A _R. H. Hardin_

%E Extended with signs by _Olivier Gérard_, Mar 15 1997