A288269 - OEIS

%I #26 Dec 01 2021 16:44:12

%S 1,0,4,36,336,3840,52800,836640,14864640,291755520,6264276480,

%T 145962432000,3665362821120,98604459233280,2827182573895680,

%U 86016204578304000,2766450467708928000,93741871082943283200,3336807307530977280000,124443669133537276723200

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

%H Seiichi Manyama, <a href="/A288269/b288269.txt">Table of n, a(n) for n = 0..420</a>

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

%F E.g.f.: exp(2*x^2/(1 - x)^3). - _Ilya Gutkovskiy_, Jul 27 2020

%F 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

%F 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

%t 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 *)

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

%o (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

%o (Magma)

%o R<x>:=PowerSeriesRing(Rationals(), 30);

%o Coefficients(R!(Laplace( Exp(2*x^2/(1-x)^3) ))); // _G. C. Greubel_, Mar 10 2021

%Y Cf. A052887, A288268.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Oct 20 2017