A130744 a(n) = n*(n+2)*n!.

0, 3, 16, 90, 576, 4200, 34560, 317520, 3225600, 35925120, 435456000, 5708102400, 80472268800, 1214269056000, 19527937228800, 333456963840000, 6025763487744000, 114887039275008000, 2304854534062080000

Offset: 0

Author

Paul Curtz, Jul 12 2007

Keywords

• nonn
• easy

Comments

For n >= 1, a(n) = number whose factorial base representation (A007623) begins with a double digit {n}{n}, which is followed by n-1 zeros. Viewed in that base, this sequence looks like this: 0, 11, 220, 3300, 44000, 550000, 6600000, 77000000, 880000000, 9900000000, AA000000000, BB0000000000, ... (where "digits" A and B stand for placeholder values 10 and 11 respectively). - Antti Karttunen, May 07 2015

Examples

			G.f. = 3*x + 16*x^2 + 90*x^3 + 576*x^4 + 4200*x^5 + 34560*x^6 + ...
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..300
• Index entries for sequences related to factorial base representation.

Crossrefs

Cf. A000142, A001048, A005563, A007623.

Column 3 of A257503 (apart from initial zero. Equally, row 3 of A257505).

Subsequence of both A227130 and A227148.

Cf. A001113, A001620, A091725, A099285.

Programs

Magma

[n*(n+2)*Factorial(n): n in [0..25]]; // Vincenzo Librandi, Aug 11 2011

Mathematica

a[n_]:=(n+2)!-(n+1)!-n!; (* Vladimir Joseph Stephan Orlovsky, Dec 05 2008 *)

PARI

a(n)=n!*(n*(n+2)) \\ Charles R Greathouse IV, Aug 11 2011

Scheme

(define (A130744 n) (* n (+ 2 n) (A000142 n))) ;; Antti Karttunen, May 07 2015

Formula

0 = +a(n) * (+a(n+1) + 2*a(n+2) - 6*a(n+3) + a(n+4)) + a(n+1) * (+5*a(n+2) - 6*a(n+3) + a(n+4)) + a(n+2) * (+3*a(n+2) - a(n+4)) + a(n+3) * (+a(n+3)) if n>=0. - Michael Somos, Mar 26 2014

From Antti Karttunen, May 07 2015: (Start)

a(n) = n * (n! + (n+1)!) = n * A001048(n+1).

a(n) = A005563(n) * A000142(n).

a(n) = (n+2)! - (n+1)! - n! [from Orlovsky's Mathematica-code].

(End)

From Amiram Eldar, May 17 2022: (Start)

Sum_{n>=1} 1/a(n) = (Ei(1) - gamma)/2 - 1/4, where Ei(1) = A091725 and gamma = A001620.

Sum_{n>=1} (-1)^(n+1)/a(n) = (gamma - Ei(-1))/2 - 1/e + 1/4, where Ei(-1) = -A099285 and e = A001113. (End)

Extensions

More terms from Vladimir Joseph Stephan Orlovsky, Dec 05 2008