A346920 Expansion of e.g.f. 1 / (1 - (exp(x) - 1)^5 / 5!).

1, 0, 0, 0, 0, 1, 15, 140, 1050, 6951, 42777, 260590, 1809060, 17418401, 229768539, 3402511476, 50013258750, 706670789371, 9659104177101, 130958047050698, 1834295186003784, 27849428308615221, 472297857494304303, 8856291348143365456, 176841068643273207426

Offset: 0

Author

Ilya Gutkovskiy, Aug 07 2021

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..469

Crossrefs

Cf. A000670, A330047, A346894, A346895.

Cf. A000481, A327506, A346924.

Programs

Mathematica

nmax = 24; CoefficientList[Series[1/(1 - (Exp[x] - 1)^5/5!), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] StirlingS2[k, 5] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 24}]

PARI

my(x='x+O('x^25)); Vec(serlaplace(1/(1-(exp(x)-1)^5/5!))) \\ Michel Marcus, Aug 07 2021

PARI

my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, (5*k)!*x^(5*k)/(120^k*prod(j=1, 5*k, 1-j*x)))) \\ Seiichi Manyama, May 09 2022

PARI

a(n) = sum(k=0, n\5, (5*k)!*stirling(n, 5*k, 2)/120^k); \\ Seiichi Manyama, May 09 2022

Formula

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * Stirling2(k,5) * a(n-k).

a(n) ~ n! / (5*(1 + 120^(-1/5)) * log(1 + 120^(1/5))^(n+1)). - Vaclav Kotesovec, Aug 08 2021

From Seiichi Manyama, May 09 2022: (Start)

G.f.: Sum_{k>=0} (5*k)! * x^(5*k)/(120^k * Product_{j=1..5*k} (1 - j * x)).

a(n) = Sum_{k=0..floor(n/5)} (5*k)! * Stirling2(n,5*k)/120^k. (End)