cp's OEIS Frontend

A152689 Apply partial sum operator thrice to factorials.

%I A152689 #14 Sep 08 2022 08:45:39
%S A152689 0,0,0,1,4,11,28,79,284,1363,8356,61583,523924,5024179,53479148,
%T A152689 624890431,7946278828,109195935539,1612048228564,25439293045903,
%U A152689 427278358483556,7609502950269523,143217213477235804,2840152418116022399
%N A152689 Apply partial sum operator thrice to factorials.
%H A152689 G. C. Greubel, <a href="/A152689/b152689.txt">Table of n, a(n) for n = 0..450</a>
%H A152689 G. V. Milovanovich and A. Petojevich, <a href="http://www.math.bas.bg/infres/MathBalk/MB-16/MB-16-113-130.pdf">Generalized Factorial Functions, Numbers and Polynomials</a>, Math. Balkanica, Vol. 16 (2002), Fasc. 1-4.
%F A152689 From _G. C. Greubel_, Sep 13 2018: (Start)
%F A152689 a(n) = Sum_{j=0..(n-1)} Sum_{m=0..(j-1)} !m, where !n = Sum_{k=0..(n-1)} k! = A003422(n).
%F A152689 a(n) = ((n^2 -3*n +1) * !n - (n-2)*n! + 2*(n-1))/2.
%F A152689 a(n) = Sum_{j=0..(n-1)} ((j-1) * !j - j! + 1) = Sum_{j=0..(n-1)} A014144(n). (End)
%t A152689 With S[n_]:= Sum[k!, {k,0,n-1}];
%t A152689 Table[Sum[Sum[S[j], {j,0,m-1}], {m,0,n -1}], {n, 0, 30}] (* or *) Table[((n^2 - 3*n + 1)*S[n] - (n - 2)*n! + 2*(n - 1))/2, {n, 0, 30}] (* _G. C. Greubel_, Sep 13 2018 *)
%o A152689 (PARI) for(n=0,30, print1(((n^2-3*n+1)*sum(k=0,n-1, k!) - (n-2)*n! + 2*(n -1))/2, ", ")) \\ _G. C. Greubel_, Sep 13 2018
%o A152689 (Magma) [0] cat [((n^2 -3*n +1)*(&+[Factorial(k): k in [0..(n-1)]]) -(n-2)*Factorial(n) +2*(n-1))/2: n in [1..30]]; // _G. C. Greubel_, Sep 13 2018
%Y A152689 Cf. A152687, A014144.
%K A152689 nonn
%O A152689 0,5
%A A152689 _Vladimir Joseph Stephan Orlovsky_, Dec 10 2008
%E A152689 Prepended zeros and changed offset by _G. C. Greubel_, Sep 13 2018