A288269 Expansion of e.g.f.: exp(Sum_{k>=1} (k-1)*k*x^k).

1, 0, 4, 36, 336, 3840, 52800, 836640, 14864640, 291755520, 6264276480, 145962432000, 3665362821120, 98604459233280, 2827182573895680, 86016204578304000, 2766450467708928000, 93741871082943283200, 3336807307530977280000, 124443669133537276723200

Offset: 0

Author

Seiichi Manyama, Oct 20 2017

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..420

Crossrefs

Cf. A052887, A288268.

Programs

Magma

R:=PowerSeriesRing(Rationals(), 30);
Coefficients(R!(Laplace( Exp(2*x^2/(1-x)^3) ))); // G. C. Greubel, Mar 10 2021

Mathematica

With[{m = 30}, CoefficientList[Series[Exp[2*x^2/(1-x)^3], {x, 0, m}], x]*Range[0, m]!] (* G. C. Greubel, Mar 10 2021 *)

PARI

{a(n) = n!*polcoeff(exp(sum(k=1, n, (k-1)*k*x^k)+x*O(x^n)), n)}

Sage

[factorial(n)*( exp(2*x^2/(1-x)^3) ).series(x,n+1).list()[n] for n in (0..30)] # G. C. Greubel, Mar 10 2021

Formula

a(0) = 1 and a(n) = (n-1)! * Sum_{k=1..n} (k-1)*k^2*a(n-k)/(n-k)! for n > 0.

E.g.f.: exp(2*x^2/(1 - x)^3). - Ilya Gutkovskiy, Jul 27 2020

a(n) ~ 3^(1/8) * exp(2/27 - (n/6)^(1/4)/12 - (n/6)^(1/2) + 8*(n/6)^(3/4) - n) * n^(n - 1/8) / 2^(7/8) * (1 - 3203/34560 * (6/n)^(1/4)). - Vaclav Kotesovec, Mar 10 2021

a(n) = 4*(n-1)*a(n-1) - 2*(n-1)*(3*n-8)*a(n-2) + 2*(n-2)*(n-1)*(2*n-5)*a(n-3) - (n-4)*(n-3)*(n-2)*(n-1)*a(n-4). - Vaclav Kotesovec, Dec 01 2021