%I #11 Dec 05 2021 10:41:44
%S 1,2,6435,4154246671960,5397234129638871133346507775,
%T 80240648651400365471854502514501453704175376562496,
%U 54198670627270688013781273396239242514947489935351300645194042280183395324517200
%N a(n) = binomial(n^3 + 3*n^2 - 3*n + 1, n^3).
%C a(n) is a sharp upper bound of the number of vertices of the polytope of the n X n X n stochastic tensors, or equivalently, of the number of Latin squares of order n, or equivalently, of the number of n X n X n line-stochastic (0,1)-tensors (see Zhang et al.).
%H Fuzhen Zhang and Xiao-Dong Zhang, <a href="https://doi.org/10.1080/02331934.2019.1647198">Enumerating extreme points of the polytopes of stochastic tensors: an optimization approach</a>, Optimization, 69:4, 729-741, (2020). <a href="https://arxiv.org/abs/2008.04655">arXiv:2008.04655 [math.CO]</a>, 2020. See p. 6.
%H Fuzhen Zhang and Xiao-Dong Zhang, <a href="https://arxiv.org/abs/2110.12337">Comparison of the upper bounds for the extreme points of the polytopes of line-stochastic tensors</a>, arXiv:2110.12337 [math.CO], 2021. See p. 5.
%F A349508(n)/A349509(n) <= A349510(n) < A349511(n) < a(n) (see Corollary 7 in Zhang et al., 2021).
%F a(n) ~ C*3^(3(n - n^2))*exp(3*(3*n/2 + n^2))*n^(3(-n + n^2)), where C = e^(-15)/sqrt(54*Pi).
%t a[n_]:=Binomial[n^3+3n^2-3n+1,n^3]; Array[a,8,0]
%Y Cf. A000578, A229013.
%Y Cf. A349506, A349507, A349508, A349509, A349510, A349511.
%K nonn
%O 0,2
%A _Stefano Spezia_, Nov 20 2021