A346894 Expansion of e.g.f. 1 / (1 - (exp(x) - 1)^3 / 3!).

1, 0, 0, 1, 6, 25, 110, 721, 6286, 57625, 541470, 5558641, 64351166, 819480025, 11140978030, 160711583761, 2472834185646, 40597082635225, 706816137889790, 12974021811748081, 250395124862965726, 5074637684604691225, 107798916619788396750

Offset

0,5

Links

Seiichi Manyama, Table of n, a(n) for n = 0..452

Formula

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * Stirling2(k,3) * a(n-k).

a(n) ~ n! / (3*(1 + 6^(-1/3)) * log(1 + 6^(1/3))^(n+1)). - Vaclav Kotesovec, Aug 08 2021

From Seiichi Manyama, May 07 2022: (Start)

G.f.: Sum_{k>=0} (3*k)! * x^(3*k)/(6^k * Product_{j=1..3*k} (1 - j * x)).

a(n) = Sum_{k=0..floor(n/3)} (3*k)! * Stirling2(n,3*k)/6^k. (End)

Mathematica

nmax = 22; CoefficientList[Series[1/(1 - (Exp[x] - 1)^3/3!), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] StirlingS2[k, 3] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 22}]

Prog

(PARI) my(x='x+O('x^25)); Vec(serlaplace(1/(1-(exp(x)-1)^3/3!))) \\ Michel Marcus, Aug 06 2021
(PARI) my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, (3*k)!*x^(3*k)/(6^k*prod(j=1, 3*k, 1-j*x)))) \\ Seiichi Manyama, May 07 2022
(PARI) a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, binomial(i, j)*stirling(j, 3, 2)*v[i-j+1])); v; \\ Seiichi Manyama, May 07 2022
(PARI) a(n) = sum(k=0, n\3, (3*k)!*stirling(n, 3*k, 2)/6^k); \\ Seiichi Manyama, May 07 2022

Crossrefs

Cf. A000670, A330047, A346895, A346920.

Cf. A000392, A327504, A346922, A353774.

Sequence in context: A212258 A048875 A295202 * A094669 A100296 A346818

Adjacent sequences: A346891 A346892 A346893 * A346895 A346896 A346897

Keyword

nonn

Author

Ilya Gutkovskiy, Aug 06 2021

Status

approved