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A293050
Expansion of e.g.f. exp(x^4/(1 - x)).
4
1, 0, 0, 0, 24, 120, 720, 5040, 60480, 725760, 9072000, 119750400, 1756339200, 28021593600, 479480601600, 8717829120000, 168254102016000, 3438311804928000, 74160828758016000, 1682757222322176000, 40061786401308672000, 998402161605488640000
OFFSET
0,5
LINKS
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)
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 *)
PROG
(PARI) x='x+O('x^66); Vec(serlaplace(exp(x^4/(1-x))))
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).
Sequence in context: A050213 A124657 A342856 * A052581 A052605 A195917
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Sep 29 2017
STATUS
approved