cp's OEIS Frontend

A191908 a(n) = Sum_{k=0..n} (k+1)^(n-1)*k!*StirlingS2(n,k).

%I A191908 #24 Jun 14 2024 11:40:49
%S A191908 1,1,8,154,5690,346366,31540898,4022618734,685081183970,
%T A191908 150294263931406,41295554517419138,13894282169096540014,
%U A191908 5619799582929595762850,2690722557848361804976846,1505284957795131345177533378,973008827731313629949682056494
%N A191908 a(n) = Sum_{k=0..n} (k+1)^(n-1)*k!*StirlingS2(n,k).
%H A191908 Vincenzo Librandi, <a href="/A191908/b191908.txt">Table of n, a(n) for n = 0..200</a>
%F A191908 a(n) = Sum_{j=0..n} ((j+1)^(n-1) * Sum_{i=0..j} (-1)^i*C(j, i)*(j-i)^n).
%F A191908 Limit n->infinity a(n)^(1/n)/n^2 = 0.42780682830692054814273... = r/exp(1) * (1/r+LambertW(-exp(-1/r)/r))^(r-1) / (-LambertW(-exp(-1/r)/r))^r, where r = 0.87370243323966833... is the root of the equation r*(1+exp(-1/r)) * LambertW(-exp(-1/r)/r) = -1.
%F A191908 E.g.f.: Sum_{n>=0} (exp((n+1)*x) - 1)^n / (n+1). - _Paul D. Hanna_, Dec 30 2012
%F A191908 O.g.f.: Sum_{n>=0} (n+1)^(n-1) * n! * x^n / Product_{k=1..n} (1 - (n+1)*k*x). - _Paul D. Hanna_, Oct 26 2014
%t A191908 Table[Sum[(k+1)^(n-1)*k!*StirlingS2[n, k],{k,0,n}],{n,0,20}]
%t A191908 Flatten[{1,Table[Sum[(j+1)^(n-1)*Sum[(-1)^i*Binomial[j,i]*(j-i)^n,{i,0,j}],{j,0,n}],{n,1,20}]}]
%t A191908 Table[n!*SeriesCoefficient[Sum[(E^(x*(k+1))-1)^k/(k+1),{k,0,n}],{x,0,n}],{n,0,20}]
%t A191908 (* program for numerical value of the limit n->infinity a(n)^(1/n)/n^2 *) r^2*Exp[1/r-2]*(1+Exp[-1/r])/.FindRoot[r*(1+Exp[-1/r])*LambertW[-Exp[-1/r]/r] == -1, {r, 1/2}, WorkingPrecision -> 50]
%o A191908 (PARI) {a(n)=n!*polcoeff(sum(k=0, n, (exp((k+1)*x+x*O(x^n)) - 1)^k/(k+1)), n)}
%o A191908 for(n=0, 20, print1(a(n), ", "))
%o A191908 (PARI) {a(n)=polcoeff(sum(m=0, n, (m+1)^(m-1)*m!*x^m/prod(k=1, m, 1-(m+1)*k*x+x*O(x^n))), n)}
%o A191908 for(n=0, 20, print1(a(n), ", ")) \\ _Paul D. Hanna_, Oct 26 2014
%Y A191908 Cf. A188634.
%K A191908 nonn
%O A191908 0,3
%A A191908 _Vaclav Kotesovec_, Dec 30 2012