A105218 a(n) = Sum_{k=0..n} C(n,k)^2*(n-k)!*k^3.

0, 1, 12, 117, 1168, 12525, 145836, 1844017, 25238592, 372320793, 5894109100, 99712850061, 1795703361552, 34303011804997, 692863434782988, 14754105717057225, 330351159979499776, 7758672154410196017, 190717243734190845132, 4896738903385469500453

Offset

0,3

Comments

Let b(n) denote the cubes (A000578). If e.g.f. of b(n) is E(x) and a(n) = Sum{k=0..n} C(n,k)^2*(n-k)!*b(k), then e.g.f. of a(n) is E(x/(1-x))/(1-x). (Thanks to Vladeta Jovovic for help.) - Miklos Kristof, Apr 19 2005

Links

Seiichi Manyama, Table of n, a(n) for n = 0..442

Formula

E.g.f. = (x/(1-x)^2+3*x^2/(1-x)^3+x^3/(1-x)^4)*exp(x/(1-x)) - Miklos Kristof, Apr 19 2005

Recurrence: (n-2)*(n-1)^2*a(n) = (n-2)*n^2*(2*n-1)*a(n-1) - (n-1)^3*n^2*a(n-2). - Vaclav Kotesovec, Sep 26 2013

a(n) ~ n^(n+7/4)*exp(2*sqrt(n)-n-1/2)/sqrt(2) * (1 - 5/(48*sqrt(n))). - Vaclav Kotesovec, Sep 26 2013

a(n) = n*n!*hypergeom([2, 1-n], [1, 1], -1). - Peter Luschny, Apr 01 2015

Example

b(n) = 0, 1, 8, 27, 64, 125, 216, ...
a(3) = C(3,0)^2*3!*b(0) + C(3,1)^2*2!*b(1) + C(3,2)^2*1!*b(2) + C(3,3)^2*0!*b(3) = 1*6*0 + 9*2*1 + 9*1*8 + 1*1*27 = 0 + 18 + 72 + 27 = 117.

Maple

seq(add(binomial(n, k)^2*(n-k)!*k^3, k=0..n), n=0..30);
# Alternatively:
a := n -> n*n!*hypergeom([2, 1-n], [1, 1], -1):
seq(simplify(a(n)), n=0..19); # Peter Luschny, Apr 01 2015

Mathematica

CoefficientList[Series[(x/(1-x)^2+3*x^2/(1-x)^3+x^3/(1-x)^4)*E^(x/(1-x)), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec after Miklos Kristof, Sep 26 2013 *)

Prog

(PARI) my(N=20, x='x+O('x^N)); concat(0, Vec(serlaplace(x*(1+x-x^2)*exp(x/(1-x))/(1-x)^4))) \\ Seiichi Manyama, Feb 06 2021

Crossrefs

Cf. A000578.

Sequence in context: A182671 A154346 A016142 * A180777 A163950 A025132

Adjacent sequences: A105215 A105216 A105217 * A105219 A105220 A105221

Keyword

easy,nonn

Author

Miklos Kristof, Apr 13 2005

Status

approved