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%I #14 Jan 29 2018 02:52:54
%S 1,2,24,552,28032,1778688,212383872,25215328512,5734229114880,
%T 1029078328135680,410202091438571520,93624495716395745280,
%U 65377722614151010222080,15441784659337549573324800
%N Expansion of e.g.f. tan(arcsin(arcsinh(x))) (odd powers only).
%F a(n) = ((2*n+1)!*sum(m=0..n, binomial(2*m,m)*2^(-2*m)*(2*m+1)!*(sum(j=0..n-m, (-1)^(n-m+j)*(sum(i=0..2*j,(2^i*Stirling1(2*m+1+i,2*m+1)* binomial(2*m+2*j,2*m+i))/(2*m+1+i)!)) *binomial(n-1/2,n-m-j))))). - _Vladimir Kruchinin_, Jun 17 2011
%e tan(arcsin(arcsinh(x))) = x + (2/3!)*x^3 + (24/5!)*x^5 + (552/7!)*x^7 + (28032/9!)*x^9 + ...
%t With[{nn=30},Take[CoefficientList[Series[Tan[ArcSin[ArcSinh[x]]],{x,0,nn}],x] Range[0,nn-1]!,{2,-1,2}]] (* _Harvey P. Dale_, Oct 09 2012 *)
%o (Maxima)
%o a(n):=((2*n+1)!*sum(binomial(2*m,m)*2^(-2*m)*(2*m+1)!*(sum((-1)^(n-m+j)*(sum((2^i*stirling1(2*m+1+i,2*m+1)*binomial(2*m+2*j,2*m+i))/(2*m+1+i)!,i,0,2*j))*binomial(n-1/2,n-m-j),j,0,n-m)),m,0,n)); /* _Vladimir Kruchinin_, Jun 17 2011 */
%K nonn
%O 1,2
%A Patrick Demichel (patrick.demichel(AT)hp.com)
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