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A296165 a(n) is the number of unimodular triangulations of [0,2]x[0,n].

%I A296165 #33 Aug 27 2025 01:01:53
%S A296165 6,64,852,12170,182132,2801708,43936824,698607816,11224598424,
%T A296165 181815529916,2964167665340,48580814410080,799696199314500,
%U A296165 13212398835196240,218976668040908248,3639020246503687098,60616163842958990268,1011775545312594580868,16918718677672553292440,283368129709983000763876
%N A296165 a(n) is the number of unimodular triangulations of [0,2]x[0,n].
%C A296165 As stated by Kaibel and Ziegler, the number of unimodular triangulations of [0,1]x[0,n] is (2n)!/(n!*n!). This gives a(1)=6.
%C A296165 No formula for a(n) is known. Aichholzer computed a(n) for n<=15.
%C A296165 Kaibel and Ziegler computed a(n) for n<=375. Aichholzer also computed the number of unimodular triangulations of [0,m]x[0,n] for m=3,4,5 and various n, and Kaibel-Ziegler extended these calculations to m=6.
%D A296165 V. Kaibel and G. Ziegler, "Counting lattice triangulations", London Math. Soc. Lecture Notes Series, Vol. 307, pp. 277-307, 2003.
%H A296165 Robin Visser, <a href="/A296165/b296165.txt">Table of n, a(n) for n = 1..800</a>
%H A296165 O. Aichholzer, <a href="http://www.ist.tugraz.at/aichholzer/research/rp/triangulations/counting/">Counting Triangulations - Olympics</a>, 2006.
%H A296165 V. Kaibel and G. M. Ziegler, <a href="https://arxiv.org/abs/math/0211268">Counting Lattice Triangulations</a>, arXiv:math/0211268 [math.CO], 2002.
%o A296165 (SageMath) # Implements the recursive formula by V. Kaibel and G. Ziegler
%o A296165 def a(n):
%o A296165     g, ans = [[0 for i in range(n+1)] for j in range(n+1)], 0
%o A296165     for (A,B) in ((x,y) for x in range(n+1) for y in range(n+1)):
%o A296165         g[A][B] = binomial((3*A+B-1)/2, A)*binomial((A+3*B-1)/2, B)
%o A296165         for (a,b) in ((x,y) for x in range(A+1) for y in range(B+1)):
%o A296165             if ((a+b)%2==1) and (a+b < A+B):
%o A296165                 g[A][B]+=g[a][b]*binomial((3*A+B-3*a-b)/2-1,A-a)*binomial((A+3*B-a-3*b)/2-1,B-b)
%o A296165     for (B,A) in ((x, y) for x in range(n+1) for y in range(x)):
%o A296165         if ((A+B)%2==1):
%o A296165             ans+=g[A][B]*binomial(2*n-(3*A+B+1)/2,n-A)*binomial(2*n-(A+3*B+1)/2,n-B)
%o A296165     return binomial(2*n, n)^2 + 2*ans  # _Robin Visser_, May 29 2025
%Y A296165 Second column of array A082640.
%K A296165 nonn,changed
%O A296165 1,1
%A A296165 _John Kieffer_, Dec 06 2017
%E A296165 More terms from _Robin Visser_, May 29 2025