A355285 Expansion of e.g.f. 1 / (1 + x + x^2/2 + x^3/3 + log(1 - x)).

1, 0, 0, 0, 6, 24, 120, 720, 7560, 76608, 810432, 9141120, 118015920, 1666336320, 25211774016, 404932155264, 6951992261760, 127203705538560, 2467434718218240, 50477473338494976, 1086707769452699904, 24573149993692615680, 582367494447600583680, 14430857455114783119360

Offset: 0

Author

Ilya Gutkovskiy, Jun 26 2022

Keywords

• nonn

Crossrefs

Cf. A007840, A047865, A226226, A232475, A355284.

Programs

Mathematica

nmax = 23; CoefficientList[Series[1/(1 + x + x^2/2 + x^3/3 + Log[1 - x]), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] (k - 1)! a[n - k], {k, 4, n}]; Table[a[n], {n, 0, 23}]

PARI

my(x='x+O('x^30)); Vec(serlaplace(1/(1 + x + x^2/2 + x^3/3 + log(1 - x)))) \\ Michel Marcus, Jun 27 2022

Formula

E.g.f.: 1 / (1 - Sum_{k>=4} x^k/k).

a(0) = 1; a(n) = Sum_{k=4..n} binomial(n,k) * (k-1)! * a(n-k).