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A349511 a(n) = Sum_{k=n^2..3*n^2-3*n+1} binomial(n^3, k).

%I A349511 #11 Dec 05 2021 10:41:33
%S A349511 1,1,162,129426405,16891063036609237658,
%T A349511 18250180714636047151855346313907038815,
%U A349511 1291091703201646062849529792547495285890126156377393082996087554,15934719293558661243731879701489946881532638280926268234547722632676376681552065231576737967805230
%N A349511 a(n) = Sum_{k=n^2..3*n^2-3*n+1} binomial(n^3, k).
%C A349511 a(n) is an 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 A349511 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 A349511 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. 4.
%F A349511 A349508(n)/A349509(n) <= A319510(n) < a(n) < A349512(n) (see Corollary 7 in Zhang et al., 2021).
%F A349511 a(n) = binomial(n^3, n^2)*2F1([1, n^2-n^3], [1+n^2], -1) - binomial(n^3, 2-3*n+3*n^2)*2F1([1, 2-3*n+3*n^2-n^3], [3(1-n+n^2)], -1), where 2F1 is the hypergeometric function.
%F A349511 a(n) ~ exp(3*n^2 - 9*n/2 + 3) * n^(3*n*(n-1)) / (sqrt(2*Pi) * 3^(3*n^2 - 3*n + 3/2)). - _Vaclav Kotesovec_, Dec 05 2021
%t A349511 a[n_]:=Sum[Binomial[n^3,k],{k,n^2,3n^2-3n+1}]; Array[a,8,0]
%Y A349511 Cf. A000290, A000578.
%Y A349511 Cf. A349506, A349507, A349508, A349509, A349510, A349512.
%K A349511 nonn
%O A349511 0,3
%A A349511 _Stefano Spezia_, Nov 20 2021