A162695 - OEIS

A162695

E.g.f. satisfies A(x) = exp( x*A(x) * exp(x*A(x)) ).

1, 1, 5, 43, 549, 9341, 199303, 5122503, 154174121, 5321093689, 207228932811, 8991136486619, 430126003707997, 22494400020052533, 1276807091011902479, 78178242047074260751, 5136433584083525179857, 360458257425576984629873

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OFFSET

0,3

LINKS

Seiichi Manyama, Table of n, a(n) for n = 0..361

R Lorentz, S Tringali, CH Yan, Generalized Goncarov polynomials, arXiv preprint arXiv:1511.04039, 2015

FORMULA

a(n) = Sum_{k=0..n} binomial(n,k) * (n+1)^(k-1) * k^(n-k).

Let A(x)^m = Sum_{n>=0} a(n,m)*x^n/n! with a(0,m)=1, then

a(n,m) = Sum_{k=0..n} binomial(n,k) * m*(n+m)^(k-1) * k^(n-k).

...

Let log(A(x)) = x*A(x)*exp(x*A(x)) = Sum_{n>=1} L(n)*x^n/n!, then

L(n) = Sum_{k=0..n} binomial(n,k) * n^(k-1) * k^(n-k) where

L(n) = n*A055779(n), where A055779(n) is the number of fat trees on n labeled vertices.

...

a(n) ~ s*sqrt((1+r*s)/(1+r*s*(3+r*s))) * n^(n-1) / (exp(n)*r^n), where r = 0.2222181377976171017... and s = 1.998622764215824983... are roots of the system of equations exp(r*s)*r*s*(1+r*s) = 1, exp(exp(r*s)*r*s) = s. - Vaclav Kotesovec, Jul 15 2014

EXAMPLE

E.g.f.: A(x) = 1 + x + 5*x^2/2! + 43*x^3/3! + 549*x^4/4! + 9341*x^5/5! +...

exp(x*A(x)) = 1 + x + 3*x^2/2! + 22*x^3/3! + 257*x^4/4! + 4136*x^5/5! +...

Log(A(x)) = x + 4*x^2/2! + 30*x^3/3! + 356*x^4/4! + 5780*x^5/5! +...;

compare log(A(x)) to the e.g.f. of A055779 given by:

x + 2*x^2/2! + 10*x^3/3! + 89*x^4/4! + 1156*x^5/5! +...

MATHEMATICA

Flatten[{1, Table[Sum[Binomial[n, k] * (n+1)^(k-1) * k^(n-k), {k, 0, n}], {n, 1, 20}]}] (* Vaclav Kotesovec, Jul 15 2014 *)

PROG

(PARI) a(n, m=1)=sum(k=0, n, binomial(n, k)*m*(n+m)^(k-1)*k^(n-k));

(PARI) /* Log(A(x)) = Sum_{n>=1} L(n)*x^n/n! where: */

L(n)=if(n<1, 0, sum(k=1, n, binomial(n, k)*n^(k-1)*k^(n-k)));