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%I #32 Apr 25 2024 13:32:37
%S 1,1,3,12,64,424,3358,30952,325488,3845724,50437624,727094704,
%T 11427436072,194468970904,3562501626672,69898886742000,
%U 1462459974022576,32502337621339552,764665424888545504,18985593544003151296,496110180329803750944,13609892277526894358016
%N E.g.f. satisfies: A'(x) = 1 + A(x)*exp(A(x)).
%C Compare to: G'(x) = 1 + x*exp(G(x)) holds when G(x) = -log(1-x).
%C a(n) is the number of increasing trees on vertex set [n] in which vertices of out-degree r come in r varieties for r>=1 or, more picturesquely, each non-leaf vertex has a favorite child. Proof. The special case phi(w) = 1 + w e^w of Theorem 1 in the Bergeron et al link implies that the e.g.f. for such trees satisfies the defining equation of the title. [_David Callan_, Jul 21 2014]
%H Vaclav Kotesovec, <a href="/A235129/b235129.txt">Table of n, a(n) for n = 1..380</a>
%H François Bergeron, Philippe Flajolet, and Bruno Salvy, <a href="http://algo.inria.fr/flajolet/Publications/BeFlSa92.pdf">Varieties of increasing trees</a>, Lecture Notes in Computer Science, Volume 581, 1992, pages 24-48.
%F E.g.f.: x + Integral Series_Reversion(G(x)) dx, where G(x) = Sum_{n>=1} (-1)^(n-1)*A054201(n)*x^n/n! and A054201(n) = (n-1)!*Sum_{k=1..n} k^k/k!.
%F Limit_{n->oo} (a(n)/n!)^(1/n) = 1.303391375867579164172246157... = 1/r, where r = Integral_{x=0..oo} (1/(1+x*exp(x))) dx = 0.767229259388315... - _Vaclav Kotesovec_, Feb 23 2014
%F Conjecture: a(n) = R(n-1, 0) where R(n, q) = R(n-1, q+1) + (q + 1)*Sum_{j=0..q} binomial(q, j)*R(n-1, j) for n > 0, q >= 0 with R(0, q) = [q = 0] for q >= 0. - _Mikhail Kurkov_, Dec 21 2023
%e E.g.f.: A(x) = x + x^2/2! + 3*x^3/3! + 12*x^4/4! + 64*x^5/5! + 424*x^6/6! + ...
%e Related expansions:
%e exp(A(x)) = 1 + x + 2*x^2/2! + 7*x^3/3! + 34*x^4/4! + 210*x^5/5! + 1574*x^6/6! + ...
%e Let G(x) be the series reversion of A(x)exp(A(x)), then
%e G(x) = x - 3*x^2/2! + 15*x^3/3! - 109*x^4/4! + 1061*x^5/5! - 13081*x^6/6! + 196135*x^7/7! + ... + (-1)^(n-1)*A054201(n)*x^n/n! + ...
%t Rest[CoefficientList[InverseSeries[Series[Integrate[1/(1+E^z*z),{z,0,x}],{x,0,10}],x],x]*Range[0,10]!] (* _Vaclav Kotesovec_, Feb 23 2014 *)
%o (PARI) {a(n)=local(A=x);for(i=1,n,A=intformal(1+A*exp(A+x*O(x^n))));n!*polcoeff(A,n)}
%o for(n=1,25,print1(a(n),", "))
%o (PARI) {A054201(n)=(n-1)!*sum(k=1,n,k^k/k!)}
%o {a(n)=local(A=x,G=sum(m=1,n,(-1)^(m-1)*A054201(m)*x^m/m!));A=x+intformal(serreverse(G+x*O(x^n)));n!*polcoeff(A,n)}
%o for(n=1,25,print1(a(n),", "))
%Y Cf. A054201.
%K nonn,changed
%O 1,3
%A _Paul D. Hanna_, Jan 03 2014
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