%I #18 Apr 12 2017 23:09:42
%S 1,4,84,4152,370128,51861888,10494283968,2894815734912,
%T 1043916757274880,476720372375608320,268870416029396075520,
%U 183537887154798761809920,149132786692921038502318080
%N Expansion of e.g.f: tan(arcsin(arcsin(x))).
%H G. C. Greubel, <a href="/A012065/b012065.txt">Table of n, a(n) for n = 0..215</a>
%F a(n) = ((2*n+1)!*Sum_{m=0..n} binomial(m-1/2,m)*(2*m+1)!*(Sum_{j=0..n-m} (-1)^(j)*(Sum_{i=0..2*j} (2^i*stirling1(2*m+1+i,2*m+1) *binomial(2*j+2*m,2*m+i))/(2*m+1+i)!))*binomial(n-1/2,n-j-m))))). - _Vladimir Kruchinin_, Jun 15 2011
%F E.g.f.: arcsin(x) / sqrt(1-arcsin(x)^2). - _Vaclav Kotesovec_, Feb 06 2015
%F a(n) ~ (2*n+1)! * sqrt(cos(1)) / (sqrt(Pi*n) * (sin(1))^(2*n+3/2)). - _Vaclav Kotesovec_, Feb 06 2015
%e tan(arcsin(arcsin(x))) = x + 4/3!*x^3 + 84/5!*x^5 + 4152/7!*x^7 + 370128/9!*x^9 ...
%t With[{nn=30},Take[CoefficientList[Series[Tan[ArcSin[ArcSin[x]]],{x,0,nn}],x] Range[0,nn-1]!,{2,-1,2}]] (* _Harvey P. Dale_, Oct 11 2014 *)
%o (Maxima)
%o a(n):=((2*n+1)!*sum(binomial(m-1/2,m)*(2*m+1)!*(sum((-1)^(j)*(sum((2^i*stirling1(2*m+1+i,2*m+1)*binomial(2*j+2*m,2*m+i))/(2*m+1+i)!,i,0,2*j))*binomial(n-1/2,n-j-m),j,0,n-m)),m,0,n)); /* _Vladimir Kruchinin_, Jun 15 2011 */
%o (PARI) x='x+O('x^50); v=Vec(serlaplace(asin(x) / sqrt(1-asin(x)^2))); vector(#v\2, n, v[2*n-1]) \\ _G. C. Greubel_, Apr 11 2017
%K nonn
%O 0,2
%A Patrick Demichel (patrick.demichel(AT)hp.com)