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A215188 E.g.f.: A(x) = x + sinh(A(x)^2).

%I A215188 #14 Jan 23 2014 08:27:04
%S A215188 1,2,12,120,1680,30360,672000,17599680,532224000,18248660640,
%T A215188 699512647680,29642193060480,1375922515968000,69427962935210880,
%U A215188 3783838462038835200,221509040567970355200,13862292728701236019200,923523471334492405977600,65257265823541297938432000
%N A215188 E.g.f.: A(x) = x + sinh(A(x)^2).
%F A215188 E.g.f.: Series_Reversion(x - sinh(x^2)).
%F A215188 E.g.f.: x + Sum_{n>=1} d^(n-1)/dx^(n-1) sinh(x^2)^n/n!.
%F A215188 E.g.f.: x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) (1/x)*sinh(x^2)^n/n! ).
%F A215188 a(n) ~ n^(n-1) / (sqrt(1/s + 4*s^2*sinh(s^2)) * (s-sinh(s^2))^(n-1/2) * exp(n)), where s = 0.4863332989938055129629... is the root of the equation 2*s*cosh(s^2) = 1. - _Vaclav Kotesovec_, Jan 23 2014
%e A215188 E.g.f.: A(x) = x + 2*x^2/2! + 12*x^3/3! + 120*x^4/4! + 1680*x^5/4! +...
%e A215188 where A(x - sinh(x^2)) = x and A(x) = x + sinh(A(x)^2).
%e A215188 Series expansions:
%e A215188 A(x) = x + sinh(x^2) + d/dx sinh(x^2)^2/2! + d^2/dx^2 sinh(x^2)^3/3! + d^3/dx^3 sinh(x^2)^4/4! +...
%e A215188 log(A(x)/x) = sinh(x^2)/x + d/dx (sinh(x^2)^2/x)/2! + d^2/dx^2 (sinh(x^2)^3/x)/3! + d^3/dx^3 (sinh(x^2)^4/x)/4! +...
%t A215188 Rest[CoefficientList[InverseSeries[Series[x - Sinh[x^2],{x,0,20}],x],x] * Range[0,20]!] (* _Vaclav Kotesovec_, Jan 23 2014 *)
%o A215188 (PARI) {a(n)=n!*polcoeff(serreverse(x-sinh(x^2+x^2*O(x^n))), n)}
%o A215188 for(n=1, 25, print1(a(n), ", "))
%o A215188 (PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
%o A215188 {a(n)=local(A=x); A=x+sum(m=1, n, Dx(m-1, sinh(x^2+x*O(x^n))^m)/m!); n!*polcoeff(A, n)}
%o A215188 (PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
%o A215188 {a(n)=local(A=x+x^2+x*O(x^n)); A=x*exp(sum(m=1, n, Dx(m-1, sinh(x^2+x*O(x^n))^m/x)/m!)+x*O(x^n)); n!*polcoeff(A, n)}
%Y A215188 Cf. A226758, A226759, A226760, A236357.
%K A215188 nonn
%O A215188 1,2
%A A215188 _Paul D. Hanna_, Aug 05 2012