A346751 Expansion of e.g.f. log( 1 + x^3 * exp(x) / 3! ).

0, 0, 0, 1, 4, 10, 10, -105, -1064, -6076, -16680, 129525, 2642860, 25431406, 130210444, -639438345, -26431524560, -382074099000, -3083015556624, 5641134587049, 726952330301940, 14940678486798610, 173111303303845060, 258953439321230731, -43858702741534022936

Offset: 0

Author

Ilya Gutkovskiy, Aug 01 2021

Keywords

• sign

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..471

Crossrefs

Cf. A000292, A009306, A145453, A346750, A346752, A346754.

Programs

Mathematica

nmax = 24; 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, 24}]

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