A293050 Expansion of e.g.f. exp(x^4/(1 - x)).

1, 0, 0, 0, 24, 120, 720, 5040, 60480, 725760, 9072000, 119750400, 1756339200, 28021593600, 479480601600, 8717829120000, 168254102016000, 3438311804928000, 74160828758016000, 1682757222322176000, 40061786401308672000, 998402161605488640000

Offset: 0

Author

Seiichi Manyama, Sep 29 2017

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..444

Crossrefs

Column k=3 of A293053.

E.g.f.: Product_{i>k} exp(x^i): A000262 (k=0), A052845 (k=1), A293049 (k=2), this sequence (k=3).

Cf. A361545, A361576.

Programs

Maple

a:= proc(n) option remember; `if`(n=0, 1, add(
      a(n-j)*binomial(n-1, j-1)*j!, j=4..n))
    end:
seq(a(n), n=0..23);  # Alois P. Heinz, Sep 29 2017

Mathematica

a[n_] := a[n] = If[n==0, 1, Sum[a[n-j] Binomial[n-1, j-1] j!, {j, 4, n}]];
a /@ Range[0, 23] (* Jean-François Alcover, Dec 21 2020, after Alois P. Heinz *)

PARI

x='x+O('x^66); Vec(serlaplace(exp(x^4/(1-x))))

Formula

E.g.f.: Product_{i>3} exp(x^i).

From Vaclav Kotesovec, Sep 30 2017: (Start)

a(n) = 2*(n-1)*a(n-1) - (n-2)*(n-1)*a(n-2) + 4*(n-3)*(n-2)*(n-1)*a(n-4) - 3*(n-4)*(n-3)*(n-2)*(n-1)*a(n-5).

a(n) ~ n^(n-1/4) * exp(-7/2 + 2*sqrt(n) - n) / sqrt(2).

(End)

From Seiichi Manyama, Jun 17 2024: (Start)

a(n) = n! * Sum_{k=0..floor(n/4)} binomial(n-3*k-1,n-4*k)/k!.

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