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A191908
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a(n) = Sum_{k=0..n} (k+1)^(n-1)*k!*StirlingS2(n,k).
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3
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1, 1, 8, 154, 5690, 346366, 31540898, 4022618734, 685081183970, 150294263931406, 41295554517419138, 13894282169096540014, 5619799582929595762850, 2690722557848361804976846, 1505284957795131345177533378, 973008827731313629949682056494
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OFFSET
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0,3
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LINKS
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FORMULA
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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
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MATHEMATICA
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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]
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PROG
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(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)}
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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