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A235129 E.g.f. satisfies: A'(x) = 1 + A(x)*exp(A(x)). 1
1, 1, 3, 12, 64, 424, 3358, 30952, 325488, 3845724, 50437624, 727094704, 11427436072, 194468970904, 3562501626672, 69898886742000, 1462459974022576, 32502337621339552, 764665424888545504, 18985593544003151296, 496110180329803750944, 13609892277526894358016 (list; graph; refs; listen; history; text; internal format)
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

Fran├žois Bergeron, Philippe Flajolet, 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->infinity (a(n)/n!)^(1/n) = 1.303391375867579164172246157... = 1/r, where r = integral(1/(1+x*exp(x)), x=0..infinity) = 0.767229259388315... - Vaclav Kotesovec, Feb 23 2014

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

Cf. A054201.

Sequence in context: A029851 A201720 A207557 * A222033 A302195 A196559

Adjacent sequences:  A235126 A235127 A235128 * A235130 A235131 A235132

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jan 03 2014

STATUS

approved

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Last modified May 29 15:42 EDT 2020. Contains 334702 sequences. (Running on oeis4.)