OFFSET
0,2
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..360
Eric Weisstein's World of Mathematics, Bell Number
Eric Weisstein's World of Mathematics, Laguerre Polynomial
Wikipedia, Laguerre polynomials
FORMULA
a(n) = ((-1)^n*n!/exp(1))*Sum_{k>=0} Laguerre(n,-2*n,k)/k!.
a(0) = 1; a(n) = (1/(n - 1)!)*Sum_{k=0..n} binomial(n,k)*(n + k - 1)!*Bell(n-k), where Bell() = A000110.
a(n) ~ c * n^n * 4^n / exp(n), where c = exp(exp(1/2) - 1)/sqrt(2) = 1.3527609882698012767793757868699146219161180684881726130481416807461987206887... - Vaclav Kotesovec, May 11 2021, updated Mar 18 2024
MATHEMATICA
Table[n! SeriesCoefficient[Exp[Exp[x] - 1]/(1 - x)^n, {x, 0, n}], {n, 0, 18}]
Table[(-1)^n n!/Exp[1] Sum[LaguerreL[n, -2 n, k]/k!, {k, 0, Infinity}], {n, 0, 18}]
Join[{1}, Table[1/(n - 1)! Sum[Binomial[n, k] (n + k - 1)! BellB[n - k], {k, 0, n}], {n, 18}]]
CROSSREFS
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, May 24 2018
STATUS
approved