OFFSET
0,2
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
FORMULA
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
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 *)
PROG
(Sage)
[factorial(3*n) for n in range(0, 11)] # Peter Luschny, Jun 06 2016
(Magma)
[Factorial(3*n): n in [0..15]]; // Vincenzo Librandi, Sep 24 2011
(Haskell)
a100732 = a000142 . a008585 -- Reinhard Zumkeller, Feb 19 2013
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Ralf Stephan, Dec 08 2004
STATUS
approved