A100732 a(n) = (3*n)!.

1, 6, 720, 362880, 479001600, 1307674368000, 6402373705728000, 51090942171709440000, 620448401733239439360000, 10888869450418352160768000000, 265252859812191058636308480000000

Offset: 0

Author

Ralf Stephan, Dec 08 2004

Keywords

• nonn
• easy

Comments

a(n) equals (-1)^n times the determinant of the (3n+1) X (3n+1) matrix with consecutive integers from 1 to 3n+1 along the main diagonal, consecutive integers from 2 to 3n+1 along the superdiagonal, consecutive integers from 1 to 3n along the subdiagonal, and 1's everywhere else (see Mathematica code below). - John M. Campbell, Jul 12 2011

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..100

Crossrefs

Cf. A000142, A010050, A100733, A100734, A143084, A143819, A268504.

Programs

Haskell

a100732 = a000142 . a008585  -- Reinhard Zumkeller, Feb 19 2013

Magma

[Factorial(3*n): n in [0..15]]; // Vincenzo Librandi, Sep 24 2011

Mathematica

Table[(-1)^n*Det[Array[KroneckerDelta[#1, #2]*(#1 - 1) + KroneckerDelta[#1, #2 - 1]*(#1) + KroneckerDelta[#1, #2 + 1]*(#1 - 2) + 1 &, {3*n + 1, 3*n + 1}]], {n, 0, 24}] (* John M. Campbell, Jul 12 2011 *)
(3Range[0,10])! (* Harvey P. Dale, Sep 23 2011 *)

Sage

[factorial(3*n) for n in range(0, 11)] # Peter Luschny, Jun 06 2016

Formula

a(n) = A000142(A008585(n)).

E.g.f.: 1/(1-x^3).

From Ilya Gutkovskiy, Jan 20 2017: (Start)

a(n) ~ sqrt(2*Pi)*3^(3*n+1/2)*n^(3*n+1/2)/exp(3*n).

Sum_{n>=0} 1/a(n) = (exp(3/2) + 2*cos(sqrt(3)/2))/(3*exp(1/2)) = A143819. (End)

Sum_{n>=0} (-1)^n/a(n) = (1 + 2*exp(3/2)*cos(sqrt(3)/2))/(3*e). - Amiram Eldar, Feb 14 2021

a(n) = A143084(2n,n). - Alois P. Heinz, Jul 12 2024