OFFSET
1,3
COMMENTS
Compare to: G'(x) = 1 + x*exp(G(x)) holds when G(x) = -log(1-x).
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]
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 1..380
François Bergeron, Philippe Flajolet, and Bruno Salvy, Varieties of increasing trees, Lecture Notes in Computer Science, Volume 581, 1992, pages 24-48.
FORMULA
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!.
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
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
EXAMPLE
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! + ...
Related expansions:
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! + ...
Let G(x) be the series reversion of A(x)exp(A(x)), then
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! + ...
MATHEMATICA
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 *)
PROG
(PARI) {a(n)=local(A=x); for(i=1, n, A=intformal(1+A*exp(A+x*O(x^n)))); n!*polcoeff(A, n)}
for(n=1, 25, print1(a(n), ", "))
(PARI) {A054201(n)=(n-1)!*sum(k=1, n, k^k/k!)}
{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)}
for(n=1, 25, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jan 03 2014
STATUS
approved