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a(n) = n*(n + a(n-1)) with a(0)=0.
17

%I #37 Mar 30 2020 21:56:47

%S 0,1,6,27,124,645,3906,27391,219192,1972809,19728190,217010211,

%T 2604122676,33853594957,473950329594,7109254944135,113748079106416,

%U 1933717344809361,34806912206568822,661331331924807979

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

%C Exponential convolution of factorials (A000142) and squares (A000290). - _Vladimir Reshetnikov_, Oct 07 2016

%H Seiichi Manyama, <a href="/A030297/b030297.txt">Table of n, a(n) for n = 0..449</a>

%F a(n) = A019461(2n).

%F For n>=2, a(n) = floor(2*e*n! - n - 2). - _Benoit Cloitre_, Feb 16 2003

%F a(n) = sum_{k=0...n} (n! / k!) * k^2. - _Ross La Haye_, Sep 21 2004

%F E.g.f.: x*(1+x)*exp(x)/(1-x). - _Vladeta Jovovic_, Dec 01 2004

%p 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;

%t a=0;lst={a};Do[a=(a+n)*n;AppendTo[lst, a], {n, 2*4!}];lst (* _Vladimir Joseph Stephan Orlovsky_, Dec 14 2008 *)

%t RecurrenceTable[{a[0]==0,a[n]==n(n+a[n-1])},a[n],{n,20}] (* _Harvey P. Dale_, Oct 22 2011 *)

%t 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 *)

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

%K nonn,easy

%O 0,3

%A _N. J. A. Sloane_, "Urkonsaud_admin" (miti(AT)tula.sitek.net)

%E Better description from _Henry Bottomley_, May 15 2000