A162695 - OEIS

%I #18 Jan 24 2025 11:59:57

%S 1,1,5,43,549,9341,199303,5122503,154174121,5321093689,207228932811,

%T 8991136486619,430126003707997,22494400020052533,1276807091011902479,

%U 78178242047074260751,5136433584083525179857,360458257425576984629873

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

%H Seiichi Manyama, <a href="/A162695/b162695.txt">Table of n, a(n) for n = 0..361</a>

%H R Lorentz, S Tringali, CH Yan, <a href="http://arxiv.org/abs/1511.04039">Generalized Goncarov polynomials</a>, arXiv preprint arXiv:1511.04039, 2015

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

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

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

%F ...

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

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

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

%F ...

%F 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

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

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

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

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

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

%t 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 *)

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

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

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

%Y Cf. A162659, A055779.

%Y Cf. A380425, A380426, A380427.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Jul 10 2009