A290690 - OEIS

%I #22 Jul 31 2021 07:38:55

%S 1,0,2,48,660,8640,132600,2520000,56046480,1375557120,36456769440,

%T 1041522451200,32083867126080,1061964845061120,37543201808112000,

%U 1409292653408640000,55917035430800544000,2337184142686903910400,102624865930477758067200

%N E.g.f.: exp(Sum_{k>=1} (k-1)^3*x^k).

%H Seiichi Manyama, <a href="/A290690/b290690.txt">Table of n, a(n) for n = 0..407</a>

%F a(0) = 1 and a(n) = (n-1)! * Sum_{k=1..n} (k-1)^3*k*a(n-k)/(n-k)! for n > 0.

%F From _Vaclav Kotesovec_, Jul 31 2021: (Start)

%F E.g.f.: exp(x^2*(1 + 4*x + x^2)/(1-x)^4).

%F a(n) ~ exp(65/384 - 101 * 2^(2/5) * 3^(4/5) * n^(1/5) / 1200 + 11 * 2^(4/5) * 3^(3/5) * n^(2/5) / 80 - 2^(-4/5) * 3^(2/5) * n^(3/5) + 5 * 2^(-7/5) * 3^(1/5) * n^(4/5) - n) * 2^(3/10) * 3^(1/10) * n^(n - 1/10) / sqrt(5).

%F (End)

%t nmax = 20; CoefficientList[Series[Exp[x^2*(1 + 4*x + x^2)/(1-x)^4], {x, 0, nmax}], x] * Range[0, nmax]! (* _Vaclav Kotesovec_, Jul 31 2021 *)

%o (PARI) {a(n) = n!*polcoeff(exp(sum(k=1, n, (k-1)^3*x^k)+x*O(x^n)), n)}

%Y E.g.f.: exp(Sum_{k>=1} (k-1)^m*x^k): A000262 (m=0), A052887 (m=1), A288270 (m=2), this sequence (m=3).

%Y Cf. A255819.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Oct 20 2017