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A152689 Apply partial sum operator thrice to factorials. 2
0, 0, 0, 1, 4, 11, 28, 79, 284, 1363, 8356, 61583, 523924, 5024179, 53479148, 624890431, 7946278828, 109195935539, 1612048228564, 25439293045903, 427278358483556, 7609502950269523, 143217213477235804, 2840152418116022399 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..450

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 A000604 A153876

Adjacent sequences:  A152686 A152687 A152688 * A152690 A152691 A152692

KEYWORD

nonn,changed

AUTHOR

Vladimir Joseph Stephan Orlovsky, Dec 10 2008

EXTENSIONS

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

STATUS

approved

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Last modified September 23 00:22 EDT 2018. Contains 315270 sequences. (Running on oeis4.)