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
KEYWORD
easy,nonn
AUTHOR
Miklos Kristof, Apr 13 2005
STATUS
approved