A336184 - OEIS

%I #6 Jul 11 2020 07:39:45

%S 1,9,53,466,5569,82656,1474045,30664656,729036801,19499288680,

%T 579487528861,18943592776032,675568129695601,26099852672860344,

%U 1085904530481561645,48407032164910589056,2301727955153266523521,116286277045753464506568,6220517619913795356269725

%N a(n) = n^3 + (1/n) * Sum_{k=1..n-1} binomial(n,k) * k * a(k) * (n-k)^3.

%F E.g.f.: -log(1 - exp(x) * x * (1 + 3*x + x^2)).

%F E.g.f.: -log(1 - Sum_{k>=1} k^3 * x^k / k!).

%F a(n) ~ (n-1)! / r^n, where r = 0.336491770414014560614859141224061461582454518... is the root of the equation exp(r)*r*(1 + 3*r + r^2) = 1. - _Vaclav Kotesovec_, Jul 11 2020

%t a[n_] := a[n] = n^3 + (1/n) Sum[Binomial[n, k] k a[k] (n - k)^3, {k, 1, n - 1}]; Table[a[n], {n, 1, 19}]

%t nmax = 19; CoefficientList[Series[-Log[1 - Exp[x] x (1 + 3 x + x^2)], {x, 0, nmax}], x] Range[0, nmax]! // Rest

%Y Cf. A000578, A279358, A300452, A305990, A308862, A336183.

%K nonn

%O 1,2

%A _Ilya Gutkovskiy_, Jul 10 2020