A195203 - OEIS

%I #30 Jun 16 2018 23:27:20

%S 1,0,2,6,48,440,5310,77952,1356152,27284112,623393370,15946253840,

%T 451464791052,14014830400584,473330219980982,17278004243854200,

%U 677844684489863760,28441920741699231392,1270962028978738313778,60259311813834246030048,3021271708308614076699380

%N E.g.f.: Sum_{n>=0} x*(n + x)^(n-1) * x^n/n!.

%C a(n) is the total number of leaves in all labeled forests with n nodes. Cf. A055541. - _Geoffrey Critzer_, Aug 22 2012

%H Alois P. Heinz, <a href="/A195203/b195203.txt">Table of n, a(n) for n = 0..150</a>

%F E.g.f.: exp(-x*LambertW(-x)).

%F E.g.f.: ( LambertW(-x)/(-x) )^x.

%F E.g.f.: ( Sum_{n>=0} (n + 1)^(n-1) * x^n/n! )^x.

%F E.g.f.: ( Sum_{n>=0} (n + x)^n * x^n/n! ) * (-x)/LambertW(-x). - _Paul D. Hanna_, Jun 16 2018

%F E.g.f.: LambertW(-x) / ( -x * Sum_{n>=0} (n - x)^n * x^n/n! ). - _Paul D. Hanna_, Jun 16 2018

%F a(n) = Sum_{k=0..floor(n/2)} C(n,k)*C(n-k-1,k-1)*(n-k)^(n-2*k)*k!. - _Alois P. Heinz_, Aug 22 2012

%F a(n) ~ exp(exp(-1)-1)*n^(n-1). - _Vaclav Kotesovec_, Jun 26 2013

%e E.g.f.: A(x) = 1 + 2*x^2/2! + 6*x^3/3! + 48*x^4/4! + 440*x^5/5! + ...

%e where

%e A(x) = 1 + x*(1+x)^0*x^1/1! + x*(2+x)*x^2/2! + x*(3+x)^2*x^3/3! + x*(4+x)^3*x^4/4! + ...

%e Also, A(x) = W(x)^x where W(x) = LambertW(-x)/(-x) and begins:

%e W(x) = 1 + x + 3*x^2/2! + 4^2*x^3/3! + 5^3*x^4/4! + 6^4*x^5/5! + ...

%p a:= n-> add(binomial(n, k)*binomial(n-k-1, k-1)*(n-k)^(n-2*k) *k!, k=0..n/2):

%p seq(a(n), n=0..30);  # _Alois P. Heinz_, Aug 22 2012

%t nn = 20; t = Sum[n^(n - 1) x^n/n!, {n, 1, nn}];

%t Range[0, nn]! CoefficientList[Series[Exp[x t] , {x, 0, nn}], x]  (* _Geoffrey Critzer_, Aug 22 2012 *)

%o (PARI) {a(n)=local(A=sum(k=0,n,x*(k+x)^(k-1)*x^k/k!)+x*O(x^n));n!*polcoeff(A,n)}

%o (PARI) {a(n)=local(W=sum(k=0,n,(k+1)^(k-1)*x^k/k!)+x*O(x^n));n!*polcoeff(W^x,n)}

%o (PARI) {a(n)=local(W=sum(k=0,n,(k+1)^(k-1)*x^k/k!)+x*O(x^n));n!*polcoeff(exp(x^2*W),n)}

%K nonn

%O 0,3

%A _Paul D. Hanna_, Sep 13 2011