A213167 a(n) = n! - (n-2)!.

1, 5, 22, 114, 696, 4920, 39600, 357840, 3588480, 39553920, 475372800, 6187104000, 86699289600, 1301447347200, 20835611596800, 354379753728000, 6381450915840000, 121289412980736000, 2426499634470912000

Offset: 2

Author

Olivier Gérard, Nov 02 2012

Keywords

• nonn
• easy

Comments

Row sums of A134433 starting from k=3.

a(n) = sum( (-1)^k*k*A008276(n,k), k=1..n-1).

a(n) = sum( (-1)^k*k*A054654(n,k), k=1..n-2).

For n >= 3, a(n) = number whose factorial base representation (A007623) begins with digits {n-1} and {n-2} followed by n-3 zeros. Viewed in that base, this sequence looks like this: 1, 21, 320, 4300, 54000, 650000, 7600000, 87000000, 980000000, A900000000, BA000000000, ... (where "digits" A and B stand for placeholder values 10 and 11 respectively). - Antti Karttunen, May 07 2015.

Links

• Index entries for sequences related to factorial base representation

Crossrefs

Column 4 of A257503 (apart from initial 1. Equally, row 4 of A257505).

Cf. A000142, A007623, A134433.

Cf. A008276, A094638, A008275, A130534.

Cf. A054654, A048994, A132393.

Cf. A067318.

Programs

Mathematica

Table[n! - (n - 2)!, {n, 2, 20}]
#[[3]]-#[[1]]&/@Partition[Range[0,20]!,3,1] (* Harvey P. Dale, Aug 10 2023 *)

Maxima

A213167(n):=n!-(n-2)!$
makelist(A213167(n),n,2,30); /* Martin Ettl, Nov 03 2012 */

Scheme

(define (A213167 n) (- (A000142 n) (A000142 (- n 2)))) ;; Antti Karttunen, May 07 2015

Formula

a(n) = n! - (n-2)!.

G.f.: (1/G(0) - 1 - x)/x^2 where G(k) = 1 - x/(x - 1/(x - (k+1)/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Dec 13 2012

G.f.: (1+x)/x^2*(1/Q(0)-1), where Q(k)= 1 - 2*k*x - x^2*(k + 1)^2/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 08 2013

G.f.: 2*Q(0), where Q(k)= 1 - 1/( (k+1)*(k+2) - x*(k+1)^2*(k+2)^2*(k+3)/(x*(k+1)*(k+2)*(k+3) - 1/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 08 2013