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A305919
a(n) = n! * [x^n] 1/(2 - exp(x))^n.
1
1, 1, 8, 99, 1704, 37625, 1014348, 32300359, 1186399952, 49376357109, 2296400723220, 118031059900523, 6643848377509368, 406471060412884753, 26856124898028246044, 1905791887135240982415, 144563460111417997403040, 11673024609379676114380877, 999663240630210837032231460
OFFSET
0,3
LINKS
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