OFFSET
0,2
COMMENTS
a(n) is the number of triples (A, B, C) of paths having no common vertices and using (1,0) and (0,1) as steps, where A is from (0,0) to (n,n), B is from (1,-1) to (n+1,n-1), and C is from (2,-2) to (n+2,n-2).
a(n) is the number of ways to fill a n X n grid with numbers 1, 2, 4, 8 such that each number divides the number to the right and to the top.
a(n) is the number of secondary GL(3) invariants contructed from n+2 distinct three component vectors. This number was evaluated by using the Molien-Weyl formula to compute the Hilbert series of the ring of invariants. - Jaco van Zyl, Jun 30 2025
LINKS
Paolo Xausa, Table of n, a(n) for n = 0..550
Robert de Mello Koch, Animik Ghosh, and Hendrik J. R. Van Zyl, Bosonic Fortuity in Vector Models, arXiv:2504.14181 [hep-th], 2025. See p. 9; Journal of High Energy Physics 06 (2025) 246.
Wikipedia, Lindström-Gessel-Viennot lemma.
FORMULA
From the Lindström-Gessel-Viennot lemma and using the definition from the first comment, a(n) is the determinant of the matrix:
C(2*n, n) C(2*n, n-1) C(2*n, n-2)
C(2*n, n+1) C(2*n, n) C(2*n, n-1)
C(2*n, n+2) C(2*n, n+1) C(2*n, n)
a(n) = 4*C(2*n+1,n-1)*C(2*n+1,n+2)*C(2*n,n+1)/n^3 for n >= 1.
a(n) ~ 4^(3*n+2) / (n^(9/2) * Pi^(3/2)). - Amiram Eldar, Oct 01 2025
a(n) = Product_{1 <= i, j <= n} (i + j + 2)/ (i + j - 1). - Peter Bala, Nov 18 2025
EXAMPLE
For n = 2, the triple {NNEE, NENE, ENEN} is valid, while {ENNE, NNEE, NEEN} is invalid.
MATHEMATICA
A382136[n_] := If[n == 0, 1, 4*Binomial[2*n+1, n-1]^2*Binomial[2*n, n-1]/n^3];
Array[A382136, 20, 0] (* Paolo Xausa, Jul 03 2025 *)
PROG
(PARI) a(n) = if(n==0, 1, 4*binomial(2*n+1, n-1)^2*binomial(2*n, n-1)/n^3)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Yifan Xie, Mar 27 2025
STATUS
approved