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

0,3

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

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

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 *)

Prog

(PARI) a(n)=(n-1)*round(n!/exp(1))-n!+1 \\ Charles R Greathouse IV, Feb 24 2011
(Magma) [(k-1)*(&+[Factorial(j): j in [0..k-1]]) - Factorial(k) + 1: k in [1..25]]; // G. C. Greubel, Sep 03 2018

Crossrefs

Cf. A000142, A003422, A137948.

Sequence in context: A371866 A090977 A324789 * A247183 A321139 A096358

Adjacent sequences: A014141 A014142 A014143 * A014145 A014146 A014147

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved