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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 (list; graph; refs; listen; history; text; internal format)
 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 Cf. A000670, A005649, A053492, A226513, A226515. Sequence in context: A341965 A230343 A293145 * A286841 A356440 A367445 Adjacent sequences: A305916 A305917 A305918 * A305920 A305921 A305922 KEYWORD nonn AUTHOR Ilya Gutkovskiy, Jun 14 2018 STATUS approved

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Last modified August 13 12:15 EDT 2024. Contains 375139 sequences. (Running on oeis4.)