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

1, 1, 8, 154, 5690, 346366, 31540898, 4022618734, 685081183970, 150294263931406, 41295554517419138, 13894282169096540014, 5619799582929595762850, 2690722557848361804976846, 1505284957795131345177533378, 973008827731313629949682056494

Offset: 0

Author

Vaclav Kotesovec, Dec 30 2012

Keywords

• nonn

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A188634.

Programs

Mathematica

Table[Sum[(k+1)^(n-1)*k!*StirlingS2[n, k],{k,0,n}],{n,0,20}]
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}]}]
Table[n!*SeriesCoefficient[Sum[(E^(x*(k+1))-1)^k/(k+1),{k,0,n}],{x,0,n}],{n,0,20}]
(* 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]

PARI

{a(n)=n!*polcoeff(sum(k=0, n, (exp((k+1)*x+x*O(x^n)) - 1)^k/(k+1)), n)}
for(n=0, 20, print1(a(n), ", "))

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)}
for(n=0, 20, print1(a(n), ", ")) \\ Paul D. Hanna, Oct 26 2014

Formula

a(n) = Sum_{j=0..n} ((j+1)^(n-1) * Sum_{i=0..j} (-1)^i*C(j, i)*(j-i)^n).

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.

E.g.f.: Sum_{n>=0} (exp((n+1)*x) - 1)^n / (n+1). - Paul D. Hanna, Dec 30 2012

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