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

0,5

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.

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)

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 *)

Prog

(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
(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

Crossrefs

Cf. A152687, A014144.

Sequence in context: A020964 A113067 A290890 * A217918 A360447 A000604

Adjacent sequences: A152686 A152687 A152688 * A152690 A152691 A152692

Keyword

nonn

Author

Vladimir Joseph Stephan Orlovsky, Dec 10 2008

Extensions

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

Status

approved