A383384 Expansion of e.g.f. exp(-5*x) / (1-x)^5.

1, 0, 5, 10, 105, 620, 5725, 52950, 571025, 6686200, 85871925, 1193029250, 17846277625, 285737086500, 4874590170125, 88245858436750, 1689282139310625, 34088182903910000, 723088091207873125, 16083522103093616250, 374280288623526655625, 9093957982779894737500

Offset

0,3

Links

Table of n, a(n) for n=0..21.

Formula

a(n) = n! * Sum_{k=0..n} (-5)^(n-k) * binomial(k+4,4)/(n-k)!.

a(n) = (n-1) * (a(n-1) + 5*a(n-2)) for n > 1.

E.g.f.: B(x)^5, where B(x) is the e.g.f. of A000166.

a(n) ~ sqrt(2*Pi) * n^(n + 9/2) / (24*exp(n+5)). - Vaclav Kotesovec, Apr 25 2025

Mathematica

With[{nn=30}, CoefficientList[Series[Exp[-5x]/(1-x)^5, {x, 0, nn}], x] Range[0, nn]!] (* Harvey P. Dale, Sep 04 2025 *)

Prog

(PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-5*x)/(1-x)^5))

Crossrefs

Column k=5 of A295181.

Cf. A001909, A383381, A383382, A383383.

Cf. A000166.

Sequence in context: A344140 A267042 A119137 * A048360 A357565 A251702

Adjacent sequences: A383381 A383382 A383383 * A383385 A383386 A383387

Keyword

nonn,easy

Author

Seiichi Manyama, Apr 24 2025

Status

approved