%I #35 Aug 10 2023 18:27:20

%S 1,5,22,114,696,4920,39600,357840,3588480,39553920,475372800,

%T 6187104000,86699289600,1301447347200,20835611596800,354379753728000,

%U 6381450915840000,121289412980736000,2426499634470912000

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

%C Row sums of A134433 starting from k=3.

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

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

%C 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.

%H <a href="/index/Fa#facbase">Index entries for sequences related to factorial base representation</a>

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

%F 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

%F 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

%F 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

%t Table[n! - (n - 2)!, {n, 2, 20}]

%t #[[3]]-#[[1]]&/@Partition[Range[0,20]!,3,1] (* _Harvey P. Dale_, Aug 10 2023 *)

%o (Maxima) A213167(n):=n!-(n-2)!$

%o makelist(A213167(n),n,2,30); /* _Martin Ettl_, Nov 03 2012 */

%o (Scheme) (define (A213167 n) (- (A000142 n) (A000142 (- n 2)))) ;; _Antti Karttunen_, May 07 2015

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

%Y Cf. A000142, A007623, A134433.

%Y Cf. A008276, A094638, A008275, A130534.

%Y Cf. A054654, A048994, A132393.

%Y Cf. A067318.

%K nonn,easy

%O 2,2

%A _Olivier Gérard_, Nov 02 2012