A030297 a(n) = n*(n + a(n-1)) with a(0)=0.

0, 1, 6, 27, 124, 645, 3906, 27391, 219192, 1972809, 19728190, 217010211, 2604122676, 33853594957, 473950329594, 7109254944135, 113748079106416, 1933717344809361, 34806912206568822, 661331331924807979

Offset: 0

Author

N. J. A. Sloane, "Urkonsaud_admin" (miti(AT)tula.sitek.net)

Keywords

• nonn
• easy

Comments

Exponential convolution of factorials (A000142) and squares (A000290). - Vladimir Reshetnikov, Oct 07 2016

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..449

Crossrefs

Cf. A019461-A019464, A006183, A054096, A111063.

Programs

Maple

f := proc(n) options remember; if n <= 1 then n elif n = 2 then 6 else -n*(n-2)*f(n-3)+(n-3)*n*f(n-2)+3*n*f(n-1)/(n-1); fi; end;

Mathematica

a=0;lst={a};Do[a=(a+n)*n;AppendTo[lst, a], {n, 2*4!}];lst (* Vladimir Joseph Stephan Orlovsky, Dec 14 2008 *)
RecurrenceTable[{a[0]==0,a[n]==n(n+a[n-1])},a[n],{n,20}] (* Harvey P. Dale, Oct 22 2011 *)
Round@Table[(2 E Gamma[n, 1] - 1) n, {n, 0, 20}] (* Round is equivalent to FullSimplify here, but is much faster - Vladimir Reshetnikov, Oct 07 2016 *)

Formula

a(n) = A019461(2n).

For n>=2, a(n) = floor(2*e*n! - n - 2). - Benoit Cloitre, Feb 16 2003

a(n) = sum_{k=0...n} (n! / k!) * k^2. - Ross La Haye, Sep 21 2004

E.g.f.: x*(1+x)*exp(x)/(1-x). - Vladeta Jovovic, Dec 01 2004

Extensions

Better description from Henry Bottomley, May 15 2000