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%I #21 Aug 31 2023 07:49:24
%S 1,1,64,6561,63403380965376,1000000000000,
%T 10061319724179153710638694400000000000000,
%U 9396559338406702410023114843902587890625,528450425551613768181656289451784661530463698944000000000000000000,13597557929083423616920569866317288159544321459878738801559053666747416576
%N a(n) is the numerator of n!^(2*n)/(n^n^2).
%C a(n) is the numerator of a lower bound of the number of the vertices of the polytope of stochastic semi-magic n X n X n cubes, 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 Ahmed et al. and Zhang et al.).
%H Stefano Spezia, <a href="/A349506/b349506.txt">Table of n, a(n) for n = 1..30</a>
%H Maya Mohsin Ahmed, <a href="https://arxiv.org/abs/math/0405476">Algebraic Combinatorics of Magic Squares</a>, University of California - Davis, Ph.D. Thesis, 2004; arXiv:math/0405476 [math.CO], 2004. See p. 43.
%H Maya Mohsin Ahmed, Jesús De Loera and Raymond Hemmecke, <a href="https://doi.org/10.1007/978-3-642-55566-4_2">Polyhedral Cones of Magic Cubes and Squares</a>. In: Aronov B., Basu S., Pach J., Sharir M. (eds) Discrete and Computational Geometry. Algorithms and Combinatorics, vol 25. Springer, Berlin, Heidelberg (2003). <a href="https://arxiv.org/abs/math/0201108">arXiv:math/0201108 [math.CO]</a>, 2002. See p. 3.
%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. 3.
%F a(n)/A349507(n) ~ n^(-n^2)*(exp(-n)*n^(n-1/2)*(1+12*n))^(2*n)*(Pi/72)^n.
%t Table[Numerator[n!^(2n)/(n^n^2)],{n,10}]
%o (PARI) a(n) = numerator(n!^(2*n)/n^n^2); \\ _Michel Marcus_, Nov 22 2021
%Y Cf. A000142, A000290, A002489, A005843, A185141.
%Y Cf. A349507 (denominators), A349508, A349509, A349510, A349511, A349512.
%K nonn,frac
%O 1,3
%A _Stefano Spezia_, Nov 20 2021
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