A305051 a(n) = n! * [x^n] exp(exp(x) - 1)/(1 - x)^n.

1, 2, 12, 119, 1655, 29647, 649925, 16852656, 504519916, 17124927207, 649856846635, 27262957861405, 1252893494644357, 62593349657218070, 3377648236341185084, 195782612085816693995, 12131925601060324633027, 800321307922970722566527, 55998398887720317868148977

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

Index entries for sequences related to 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

Cf. A000110, A101053, A101054, A101055.

Sequence in context: A009546 A009748 A212414 * A274305 A317013 A369075

Adjacent sequences: A305048 A305049 A305050 * A305052 A305053 A305054

Keyword

nonn

Author

Ilya Gutkovskiy, May 24 2018

Status

approved