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%I #25 Jan 28 2018 02:27:01
%S 0,2,-16,366,-17704,1467370,-185815884,33370050910,-8067253019536,
%T 2526062494781394,-994534162338738580,480859837194669214150,
%U -280103496938395910686680,193472520727526106582807226
%N Expansion of e.g.f. log(1 + sin(x)*x), even powers only.
%H Vincenzo Librandi, <a href="/A009341/b009341.txt">Table of n, a(n) for n = 0..100</a>
%F a(n) = 2*sum(k=1..2*n-1, binomial(2*n,k)*(k-1)!*(sum(i=0..k/2, (2*i-k)^(2*n-k)*binomial(k,i)*(-1)^(n-i+k-1)))/(2^k)). - _Vladimir Kruchinin_, Jun 28 2011
%F a(n) ~ (-1)^(n+1) * (2*n)! / (n*r^(2*n)), where r = 0.9320200293523439... (see A133867) is the root of the equation r*sinh(r)=1. - _Vaclav Kotesovec_, Apr 20 2014
%t With[{nn=30},Take[CoefficientList[Series[Log[1+Sin[x]x],{x,0,nn}],x] Range[0,nn]!,{1,-1,2}]] (* _Harvey P. Dale_, Nov 27 2013 *)
%o (Maxima)
%o a(n):=2*sum(binomial(2*n,k)*(k-1)!*(sum((2*i-k)^(2*n-k)*binomial(k,i)*(-1)^(n-i+k-1),i,0,k/2))/(2^k),k,1,2*n-1); /* _Vladimir Kruchinin_, Jun 28 2011 */
%Y Cf. A133867.
%K sign
%O 0,2
%A _R. H. Hardin_
%E Extended with signs by _Olivier Gérard_, Mar 15 1997
%E Previous Mathematica program replaced by _Harvey P. Dale_, Nov 27 2013
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