A382136 Number of triples of non-crossing lattice paths from (0,0) to (n,n) using (1,0) and (0,1) as steps.

1, 4, 50, 980, 24696, 731808, 24293412, 877262100, 33803832920, 1371597504992, 58043512597616, 2543610972177184, 114801908084920000, 5313688317073440000, 251370667949555421000, 12120154230252872020500, 594283640753967620247000, 29576997448419995135100000

Offset: 0

Author

Yifan Xie, Mar 27 2025

Keywords

• nonn
• easy

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

Examples

			For n = 2, the triple {NNEE, NENE, ENEN} is valid, while {ENNE, NNEE, NEEN} is invalid.
		

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, Lindstrom-Gessel-Viennot lemma

Crossrefs

Cf. A000108, A000891.

Programs

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 *)

PARI

a(n) = if(n==0, 1, 4*binomial(2*n+1, n-1)^2*binomial(2*n, n-1)/n^3)

Formula

From the Lindstrom-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.