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%I #14 Nov 08 2013 18:54:58
%S 1,-2,16,-104,-20096,4427776,-954111872,243390205696,-75389245067264,
%T 28248828019830784,-12669814369258471424,6721694045416881553408,
%U -4170436153219300846567424,2994608522937575414450814976
%N E.g.f. sinh(sin(arctan(x))) (odd powers only).
%F a(n) = ((2*n+1)!*sum(m=0..n, C(n-1/2,n-m)*(-1)^(n-m)/(2*m+1)!)). - _Vladimir Kruchinin_, Jun 16 2011
%F a(n) = -2*(6*n^2 - 6*n + 1)*a(n-1) - 12*(n-1)^2*(2*n-3)*(2*n-1)*a(n-2) - 4*(n-2)*(n-1)*(2*n-5)*(2*n-3)^2*(2*n-1)*a(n-3). - _Vaclav Kotesovec_, Nov 09 2013
%F Lim sup n->infinity |a(n)|/(2^(2*n+5/3) * exp(3/4*(2*n)^(1/3)-2*n) * n^(2*n+2/3) / sqrt(3)) = 1. - _Vaclav Kotesovec_, Nov 09 2013
%e sinh(sin(arctan(x))) = x-2/3!*x^3+16/5!*x^5-104/7!*x^7-20096/9!*x^9...
%t Table[n!*SeriesCoefficient[Sinh[x/Sqrt[1+x^2]],{x,0,n}],{n,1,41,2}] (* _Vaclav Kotesovec_, Nov 08 2013 *)
%o (Maxima)
%o a(n):=((2*n+1)!*sum(binomial(n-1/2,n-m)*(-1)^(n-m)/(2*m+1)!,m,0,n)); [_Vladimir Kruchinin_, Jun 16 2011]
%K sign
%O 1,2
%A Patrick Demichel (patrick.demichel(AT)hp.com)
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