A346754 Expansion of e.g.f. -log( 1 - x^3 * exp(x) / 3! ).

0, 0, 0, 1, 4, 10, 30, 175, 1176, 7364, 50520, 425205, 4010380, 39433966, 414654604, 4793188855, 59834495280, 789420239560, 11016095913456, 163423065359529, 2565467553034740, 42320595474149650, 732058678770177220, 13275485607004016011

Offset: 0

Author

Ilya Gutkovskiy, Aug 01 2021

Keywords

• nonn

Crossrefs

Cf. A000292, A009444, A145453, A292954, A305990, A346751, A346753, A346755.

Programs

Mathematica

nmax = 23; CoefficientList[Series[-Log[1 - x^3 Exp[x]/3!], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 0; a[n_] := a[n] = Binomial[n, 3] + (1/n) Sum[Binomial[n, k] Binomial[n - k, 3] k a[k], {k, 1, n - 1}]; Table[a[n], {n, 0, 23}]

Formula

a(0) = 0; a(n) = binomial(n,3) + (1/n) * Sum_{k=1..n-1} binomial(n,k) * binomial(n-k,3) * k * a(k).

a(n) ~ (n-1)! / (3*LambertW(2^(1/3)/3^(2/3)))^n. - Vaclav Kotesovec, Aug 08 2021

a(n) = n! * Sum_{k=1..floor(n/3)} k^(n-3*k-1)/(6^k * (n-3*k)!). - Seiichi Manyama, Dec 14 2023