A346954 Expansion of e.g.f. -log( 1 - (exp(x) - 1)^4 / 4! ).

1, 10, 65, 350, 1736, 9030, 60355, 561550, 6188996, 69919850, 781211795, 8854058850, 106994019406, 1433756147470, 21287253921635, 339206526695750, 5630710652048216, 96341917117951890, 1708973354556320875, 31787279786739738250, 623964823224788294426

Offset: 4

Author

Ilya Gutkovskiy, Aug 08 2021

Keywords

• nonn

Crossrefs

Cf. A000453, A000629, A003704, A327505, A346390, A346895, A346955.

Programs

Mathematica

nmax = 24; CoefficientList[Series[-Log[1 - (Exp[x] - 1)^4/4!], {x, 0, nmax}], x] Range[0, nmax]! // Drop[#, 4] &
a[n_] := a[n] = StirlingS2[n, 4] + (1/n) Sum[Binomial[n, k] StirlingS2[n - k, 4] k a[k], {k, 1, n - 1}]; Table[a[n], {n, 4, 24}]

Formula

a(n) = Stirling2(n,4) + (1/n) * Sum_{k=1..n-1} binomial(n,k) * Stirling2(n-k,4) * k * a(k).

a(n) ~ (n-1)! / (log(2^(3/4)*3^(1/4) + 1))^n. - Vaclav Kotesovec, Aug 09 2021

a(n) = Sum_{k=1..floor(n/4)} (4*k)! * Stirling2(n,4*k)/(k * 24^k). - Seiichi Manyama, Jan 23 2025