A009341 Expansion of e.g.f. log(1 + sin(x)*x), even powers only.

0, 2, -16, 366, -17704, 1467370, -185815884, 33370050910, -8067253019536, 2526062494781394, -994534162338738580, 480859837194669214150, -280103496938395910686680, 193472520727526106582807226

Offset: 0

Author

R. H. Hardin

Keywords

• sign

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..100

Crossrefs

Cf. A133867.

Programs

Mathematica

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 *)

Maxima

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 */

Formula

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

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

Extensions

Extended with signs by Olivier Gérard, Mar 15 1997

Previous Mathematica program replaced by Harvey P. Dale, Nov 27 2013