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%I #10 Jan 17 2018 03:16:38
%S 1,2,12,124,1800,33654,769664,20811960,649520640,22977857130,
%T 908626363392,39716055829044,1901448823572480,98954978833285470,
%U 5562010902732226560,335794917699707735664,21671616664285622599680,1488915133606275294428370,108495148653912606138040320
%N E.g.f. equals the series reversion of x - x*arcsin(x).
%F E.g.f. A(x) satisfies:
%F (1) A(x - x*arcsin(x)) = x.
%F (2) A(x) = x/(1 - arcsin(A(x))).
%F (3) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) x^n * arcsin(x)^n / n!.
%F (4) A(x) = x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(n-1) * arcsin(x)^n / n! ).
%F a(n) ~ n^(n-1) * s*sqrt((1-s^2)/(2-s^2)) / (exp(n) * (s*(1-arcsin(s)))^n), where s = 0.46154056366132202860... is the root of the equation s/(sqrt(1-s^2)*(1-arcsin(s))) = 1. - _Vaclav Kotesovec_, Jan 13 2014
%e E.g.f.: A(x) = x + 2*x^2/2! + 12*x^3/3! + 124*x^4/4! + 1800*x^5/5! + ...
%e where A(x) = x/(1 - arcsin(A(x))).
%e The e.g.f. satisfies:
%e (3) A(x) = x + x*arcsin(x) + d/dx x^2*arcsin(x)^2/2! + d^2/dx^2 x^3*arcsin(x)^3/3! + d^3/dx^3 x^4*arcsin(x)^4/4! + ...
%e (4) log(A(x)/x) = arcsin(x) + d/dx x*arcsin(x)^2/2! + d^2/dx^2 x^2*arcsin(x)^3/3! + d^3/dx^3 x^3*arcsin(x)^4/4! + ...
%t Rest[CoefficientList[InverseSeries[Series[x - x*ArcSin[x], {x, 0, 20}], x],x] * Range[0, 20]!] (* _Vaclav Kotesovec_, Jan 13 2014 *)
%o (PARI) {a(n)=n!*polcoeff(serreverse(x-x*asin(x +x*O(x^n))), n)}
%o for(n=1,25,print1(a(n),", "))
%o (PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
%o {a(n)=local(A=x); A=x+sum(m=1, n, Dx(m-1, x^m*asin(x+x*O(x^n))^m/m!)); n!*polcoeff(A, n)}
%o for(n=1,25,print1(a(n),", "))
%o (PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
%o {a(n)=local(A=x+x^2+x*O(x^n)); A=x*exp(sum(m=1, n, Dx(m-1, x^(m-1)*asin(x+x*O(x^n))^m/m!)+x*O(x^n))); n!*polcoeff(A, n)}
%o for(n=1,25,print1(a(n),", "))
%Y Cf. A227459.
%K nonn
%O 1,2
%A _Paul D. Hanna_, Jul 13 2013
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