A101055 Expansion of e.g.f. exp(exp(x)-1)/(1-x)^3.

1, 4, 20, 119, 819, 6397, 55919, 541144, 5746596, 66475311, 832418065, 11222752125, 162133146877, 2499401777680, 40960858008040, 711240364356155, 13045720176453587, 252079975222183461, 5118581045978055067, 108972887981432267708, 2427417968714846394712, 56467770394205361146187

Offset

0,2

Comments

Sequence appears in the problem of normal ordering of functions of boson operators.

Links

Harvey P. Dale, Table of n, a(n) for n = 0..447

Index entries for sequences related to Laguerre polynomials

Formula

a(n) = ((-1)^n*n!/e)*Sum_{k>=0} L(n,-n-3,k)/k!, where L is a generalized Laguerre polynomial.

a(n) = (1/2)*Sum_{k=0..n} binomial(n,k)*(k + 2)!*Bell(n-k), where Bell() = A000110. - Ilya Gutkovskiy, May 24 2018

a(n) ~ exp(exp(1)-1) * n^2 * n! / 2. - Vaclav Kotesovec, Jun 26 2022

a(0) = 1; a(n) = Sum_{k=1..n} (3*(k-1)! + 1) * binomial(n-1,k-1) * a(n-k). - Seiichi Manyama, Nov 20 2025

Mathematica

With[{nn=30}, CoefficientList[Series[Exp[Exp[x]-1]/(1-x)^3, {x, 0, nn}], x] Range[0, nn]!] (* Harvey P. Dale, Aug 11 2021 *)

Crossrefs

Cf. A000110, A101053, A101054, A305051.

Sequence in context: A128236 A390810 A091046 * A208232 A013197 A319788

Adjacent sequences: A101052 A101053 A101054 * A101056 A101057 A101058

Keyword

nonn

Author

Karol A. Penson, Nov 29 2004

Extensions

Terms after a(15) from Ilya Gutkovskiy, May 24 2018

Status

approved