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%I #6 Jan 08 2014 19:58:28
%S 1,2,13,140,2101,40502,954073,26557400,852911401,31042592042,
%T 1262704455013,56767589130980,2795116027239901,149590982933830622,
%U 8646295934108179633,536766403519254357680,35620604244949591333201
%N E.g.f. satisfies A(x) = sinh(x + A(x)^2).
%H Alois P. Heinz, <a href="/A143137/b143137.txt">Table of n, a(n) for n = 1..150</a>
%F E.g.f.: A(x) = sinh(G(x)) where G(x) = Series_Reversion( x - sinh(x)^2 ) is the e.g.f. of A143136.
%F a(n) ~ sqrt(1+sqrt(2)) * 2^(n-7/4) * n^(n-1) / (exp(n) * (1-sqrt(2)+log(1+sqrt(2)))^(n-1/2)). - _Vaclav Kotesovec_, Jan 08 2014
%e A(x) = x + 2*x^2/2! + 13*x^3/3! + 140*x^4/4! + 2101*x^5/5! +...
%e A(x) = sinh(G(x)) where G(x) is the e.g.f. of A143136:
%e G(x) = x + 2*x^2/2! + 12*x^3/3! + 128*x^4/4! + 1920*x^5/5! +...
%t Rest[CoefficientList[InverseSeries[Series[-x^2 + ArcSinh[x],{x,0,20}],x],x] * Range[0,20]!] (* _Vaclav Kotesovec_, Jan 08 2014 *)
%o (PARI) {a(n)=n!*polcoeff(sinh(serreverse(x-sinh(x+x*O(x^n))^2)),n)}
%o (PARI) {a(n)=local(A=x);for(i=0,n,A=x + sinh(A)^2);n!*polcoeff(sinh(A),n)}
%Y Cf. A143135, A143136.
%K nonn
%O 1,2
%A _Paul D. Hanna_, Jul 27 2008
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