OFFSET
0,3
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..350
N. J. A. Sloane, Transforms
FORMULA
a(n) = [x^n] Sum_{k>=0} binomial(n+k-1,k)*k!*x^k/Product_{j=1..k} (1 - j*x).
a(n) = Sum_{k=0..n} Stirling2(n,k)*binomial(n+k-1,k)*k!.
a(n) ~ n! * c * ((1 + r)*(1 + 2*r))^n / sqrt(n), where r = (-1 + 1/(-1 + LambertW(2*exp(1))))/2 = 0.833964643008471735434624869020826957702396269585... is the root of the equation (2 + 1/r) * (1 + r*LambertW(-exp(-1/r)/r)) = 1 and c = 1/sqrt(2*Pi*(1 + LambertW(2*exp(1)))) = 0.258877607195571655640738032164006... Equivalently, a(n) ~ LambertW(2*exp(1))^n * n^n / (sqrt(1 + LambertW(2*exp(1))) * 2^n * exp(n) * (LambertW(2*exp(1)) - 1)^(2*n)). - Vaclav Kotesovec, Dec 15 2019, updated Mar 17 2024
MATHEMATICA
Table[n! SeriesCoefficient[1/(2 - Exp[x])^n, {x, 0, n}], {n, 0, 18}]
Table[SeriesCoefficient[Sum[Binomial[n + k - 1, k] k! x^k/Product[1 - j x, {j, 1, k}], {k, 0, n}], {x, 0, n}], {n, 0, 18}]
Table[Sum[StirlingS2[n, k] Binomial[n + k - 1, k] k!, {k, 0, n}], {n, 0, 18}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Jun 14 2018
STATUS
approved