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
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Aug 06 2021
STATUS
approved