A014144 Apply partial sum operator twice to factorials.

0, 1, 3, 7, 17, 51, 205, 1079, 6993, 53227, 462341, 4500255, 48454969, 571411283, 7321388397, 101249656711, 1502852293025, 23827244817339, 401839065437653, 7182224591785967, 135607710526966281, 2696935204638786595, 56349204870460046909, 1234002202313888987223

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Equals row sums of triangle A137948 starting with offset 1. - Gary W. Adamson, Feb 28 2008

If s(n) is a sequence defined as s(0)=a, s(1)=b, s(n) = n*(s(n-1) - s(n-2)), n>1, then s(n) = n*b - (a(n)-1)*a. - Gary Detlefs, Feb 23 2011

Links

• G. C. Greubel, Table of n, a(n) for n = 0..250
• G. V. Milovanovich and A. Petojevich, Generalized Factorial Functions, Numbers and Polynomials, Math. Balkanica, Vol. 16 (2002), Fasc. 1-4.
• Alexsandar Petojevic, The Function vM_m(s; a; z) and Some Well-Known Sequences, Journal of Integer Sequences, Vol. 5 (2002), Article 02.1.7.
• Index entries for sequences related to factorial numbers

Crossrefs

Cf. A000142, A003422, A137948.

Programs

Magma

[(k-1)*(&+[Factorial(j): j in [0..k-1]]) - Factorial(k) + 1: k in [1..25]]; // G. C. Greubel, Sep 03 2018

Maple

b:= proc(n) option remember; `if`(n<0, [0$2],
      (q-> (f-> [f[2]+q, q]+f)(b(n-1)))(n!))
    end:
a:= n-> b(n-1)[1]:
seq(a(n), n=0..25);  # Alois P. Heinz, Feb 13 2022

Mathematica

Join[{0}, Accumulate@ Accumulate@ (Range[0, 19]!)] (* Robert G. Wilson v *)
Join[{0},Nest[Accumulate[#]&,Range[0,25]!,2]] (* Harvey P. Dale, May 12 2025 *)

PARI

a(n)=(n-1)*round(n!/exp(1))-n!+1 \\ Charles R Greathouse IV, Feb 24 2011

Formula

a(n) = (n-1) * !n - n! + 1, !n = Sum_{k=0..n-1} k!. - Joe Keane (jgk(AT)jgk.org)

a(n) = convolution(A000142, A001477). - Peter Luschny, Jan 21 2012

G.f.: x*G(0)/(1-x)^2, where G(k)= 1 + (2*k + 1)*x/( 1 - 2*x*(k+1)/(2*x*(k+1) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 24 2013