A382136 - OEIS

%I #47 Nov 20 2025 14:58:23

%S 1,4,50,980,24696,731808,24293412,877262100,33803832920,1371597504992,

%T 58043512597616,2543610972177184,114801908084920000,

%U 5313688317073440000,251370667949555421000,12120154230252872020500,594283640753967620247000,29576997448419995135100000

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

%C 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).

%C 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.

%C 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

%H Paolo Xausa, <a href="/A382136/b382136.txt">Table of n, a(n) for n = 0..550</a>

%H Robert de Mello Koch, Animik Ghosh, and Hendrik J. R. Van Zyl, <a href="https://arxiv.org/abs/2504.14181">Bosonic Fortuity in Vector Models</a>, arXiv:2504.14181 [hep-th], 2025. See p. 9; Journal of High Energy Physics 06 (2025) 246.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Lindstr%C3%B6m%E2%80%93Gessel%E2%80%93Viennot_lemma">Lindström-Gessel-Viennot lemma</a>.

%F From the Lindström-Gessel-Viennot lemma and using the definition from the first comment, a(n) is the determinant of the matrix:

%F   C(2*n, n)    C(2*n, n-1)  C(2*n, n-2)

%F   C(2*n, n+1)  C(2*n, n)    C(2*n, n-1)

%F   C(2*n, n+2)  C(2*n, n+1)  C(2*n, n)

%F 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.

%F a(n) ~ 4^(3*n+2) / (n^(9/2) * Pi^(3/2)). - _Amiram Eldar_, Oct 01 2025

%F a(n) = Product_{1 <= i, j <= n} (i + j + 2)/ (i + j - 1). - _Peter Bala_, Nov 18 2025

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

%t A382136[n_] := If[n == 0, 1, 4*Binomial[2*n+1, n-1]^2*Binomial[2*n, n-1]/n^3];

%t Array[A382136, 20, 0] (* _Paolo Xausa_, Jul 03 2025 *)

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

%Y Cf. A000108, A000891, A383823.

%K nonn,easy

%O 0,2

%A _Yifan Xie_, Mar 27 2025