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

%I A382136 #35 Jul 03 2025 02:35:40
%S A382136 1,4,50,980,24696,731808,24293412,877262100,33803832920,1371597504992,
%T A382136 58043512597616,2543610972177184,114801908084920000,
%U A382136 5313688317073440000,251370667949555421000,12120154230252872020500,594283640753967620247000,29576997448419995135100000
%N A382136 Number of triples of non-crossing lattice paths from (0,0) to (n,n) using (1,0) and (0,1) as steps.
%C A382136 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 A382136 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 A382136 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 A382136 Paolo Xausa, <a href="/A382136/b382136.txt">Table of n, a(n) for n = 0..550</a>
%H A382136 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 A382136 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lindstr%C3%B6m%E2%80%93Gessel%E2%80%93Viennot_lemma">Lindstrom-Gessel-Viennot lemma</a>
%F A382136 From the Lindstrom-Gessel-Viennot lemma and using the definition from the first comment, a(n) is the determinant of the matrix:
%F A382136   C(2*n, n)    C(2*n, n-1)  C(2*n, n-2)
%F A382136   C(2*n, n+1)  C(2*n, n)    C(2*n, n-1)
%F A382136   C(2*n, n+2)  C(2*n, n+1)  C(2*n, n)
%F A382136 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.
%e A382136 For n = 2, the triple {NNEE, NENE, ENEN} is valid, while {ENNE, NNEE, NEEN} is invalid.
%t A382136 A382136[n_] := If[n == 0, 1, 4*Binomial[2*n+1, n-1]^2*Binomial[2*n, n-1]/n^3];
%t A382136 Array[A382136, 20, 0] (* _Paolo Xausa_, Jul 03 2025 *)
%o A382136 (PARI) a(n) = if(n==0, 1, 4*binomial(2*n+1, n-1)^2*binomial(2*n, n-1)/n^3)
%Y A382136 Cf. A000108, A000891.
%K A382136 nonn,easy
%O A382136 0,2
%A A382136 _Yifan Xie_, Mar 27 2025