A293145 a(n) = n! * [x^n] exp(n*x/(1 - x)).

1, 1, 8, 99, 1696, 37225, 997056, 31535371, 1150303232, 47538819729, 2195314048000, 112032721984051, 6261138045038592, 380309520560089081, 24946892219825709056, 1757549042234670166875, 132356128415391650676736, 10610067001068927596601889, 902057202129607760380428288

Offset: 0

Author

Ilya Gutkovskiy, Oct 01 2017

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..356
• Index entries for sequences related to Laguerre polynomials

Crossrefs

Main diagonal of A253286.

Cf. A000262, A052857, A052897, A255806, A277372.

Programs

Magma

[n eq 0 select 1 else Factorial(n)*Evaluate(LaguerrePolynomial(n-1, 1), -n): n in [0..20]]; // G. C. Greubel, Feb 23 2021

Mathematica

Table[n! SeriesCoefficient[Exp[n x/(1 - x)], {x, 0, n}], {n, 0, 18}]
Table[n! SeriesCoefficient[Product[Exp[n x^k], {k, 1, n}], {x, 0, n}], {n, 0, 18}]
Join[{1}, Table[Sum[n^k n!/k! Binomial[n - 1, k - 1], {k, n}], {n, 1, 18}]]
Join[{1}, Table[n n! Hypergeometric1F1[1 - n, 2, -n], {n, 1, 18}]]
Table[If[n==0, 1, n!*LaguerreL[n-1, 1, -n]], {n, 0, 20}] (* G. C. Greubel, Feb 23 2021 *)

PARI

{a(n) = if(n==0, 1, n!*sum(k=1, n, n^k*binomial(n-1, k-1)/k!))} \\ Seiichi Manyama, Feb 03 2021

PARI

a(n) = if (n, n! * pollaguerre(n-1, 1, -n), 1); \\ Michel Marcus, Feb 23 2021

Sage

[1 if n==0 else factorial(n)*gen_laguerre(n-1, 1, -n) for n in (0..20)] # G. C. Greubel, Feb 23 2021

Formula

a(n) = n! * [x^n] Product_{k>=1} exp(n*x^k).

a(n) ~ exp(n/phi - n) * phi^(2*n) * n^n / 5^(1/4), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Oct 01 2017

a(n) = n! * Sum_{k=1..n} n^k * binomial(n-1,k-1)/k! for n > 0. - Seiichi Manyama, Feb 03 2021

a(n) = n! * LaguerreL(n-1, 1, -n) with a(0) = 1. - G. C. Greubel, Feb 23 2021