A152689 Apply partial sum operator thrice to factorials.

0, 0, 0, 1, 4, 11, 28, 79, 284, 1363, 8356, 61583, 523924, 5024179, 53479148, 624890431, 7946278828, 109195935539, 1612048228564, 25439293045903, 427278358483556, 7609502950269523, 143217213477235804, 2840152418116022399

Offset: 0

Author

Vladimir Joseph Stephan Orlovsky, Dec 10 2008

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..450
• G. V. Milovanovich and A. Petojevich, Generalized Factorial Functions, Numbers and Polynomials, Math. Balkanica, Vol. 16 (2002), Fasc. 1-4.

Crossrefs

Cf. A152687, A014144.

Programs

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

Mathematica

With S[n_]:= Sum[k!, {k,0,n-1}];
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 *)

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

Formula

From G. C. Greubel, Sep 13 2018: (Start)

a(n) = Sum_{j=0..(n-1)} Sum_{m=0..(j-1)} !m, where !n = Sum_{k=0..(n-1)} k! = A003422(n).

a(n) = ((n^2 -3*n +1) * !n - (n-2)*n! + 2*(n-1))/2.

a(n) = Sum_{j=0..(n-1)} ((j-1) * !j - j! + 1) = Sum_{j=0..(n-1)} A014144(n). (End)

Extensions

Prepended zeros and changed offset by G. C. Greubel, Sep 13 2018