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%I #26 Oct 27 2018 06:19:23
%S 1,1,3,14,91,756,7657,91504,1260441,19663280,342669691,6597811584,
%T 139094618467,3186675803584,78834061767825,2094418664339456,
%U 59474007876381553,1797637447068293376,57623116235327599411
%N E.g.f. satisfies: A'(x) = A(x)^2 + x*A(x)^3, with A(0) = 1.
%H Vaclav Kotesovec, <a href="/A183611/b183611.txt">Table of n, a(n) for n = 0..400</a>
%H V. Dotsenko, <a href="http://arxiv.org/abs/1110.0844">Pattern avoidance in labelled trees</a>, arXiv preprint arXiv:1110.0844 [math.CO], 2011-2012.
%F E.g.f.: A(x) = 1 + A(x)*[Integral 1 + x*A(x) dx], where the integration does not include the constant term.
%F E.g.f.: d/dx Series_Reversion(Sum_{n>=1} x^(3*n-2)/(3*n-2)! - x^(3*n-1)/(3*n-1)!).
%F a(n) ~ n^n * exp(Pi*(n+1)/(3*sqrt(3))-n). - _Vaclav Kotesovec_, Feb 19 2014
%e E.g.f.: A(x) = 1 + x + 3*x^2/2! + 14*x^3/3! + 91*x^4/4! +...
%e A'(x) = 1 + 3*x + 14*x^2/2! + 91*x^3/3! + 756*x^4/4! +...
%e A(x)^2 = 1 + 2*x + 8*x^2/2! + 46*x^3/3! + 348*x^4/4! + 3262*x^5/5! +...
%e A(x)^3 = 1 + 3*x + 15*x^2/2! + 102*x^3/3! + 879*x^4/4! + 4395*x^5/5! +...
%e E.g.f. A(x) = d/dx Series_Reversion(G(x)) where G(x) begins:
%e G(x) = x - x^2/2! + x^4/4! - x^5/5! + x^7/7! - x^8/8! + x^10/10! - x^11/11! +...
%e The series reversion of G(x) begins:
%e x + x^2/2! + 3*x^3/3! + 14*x^4/4! + 91*x^5/5! + 756*x^6/6! +...
%t terms = 20; A[_] = 0;
%t Do[A[x_] = 1+Integrate[A[x]^2 + x A[x]^3, x]+O[x]^terms // Normal, terms];
%t CoefficientList[A[x], x] Range[0, terms-1]! (* _Jean-François Alcover_, Oct 27 2018 *)
%o (PARI) {a(n)=local(A=1);for(n=0,n,A=1+A*intformal(1+x*A+x*O(x^n)));n!*polcoeff(A,n)}
%o (PARI) {a(n)=n!*polcoeff(deriv(serreverse(sum(m=1,n\3+1,x^(3*m-2)/(3*m-2)!-x^(3*m-1)/(3*m-1)!+x^2*O(x^n)))),n)}
%Y Cf. A199670, A049774.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Mar 21 2011
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