OFFSET
1,3
COMMENTS
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.).
LINKS
Stefano Spezia, Table of n, a(n) for n = 1..30
Maya Mohsin Ahmed, Algebraic Combinatorics of Magic Squares, University of California - Davis, Ph.D. Thesis, 2004; arXiv:math/0405476 [math.CO], 2004. See p. 43.
Maya Mohsin Ahmed, Jesús De Loera and Raymond Hemmecke, Polyhedral Cones of Magic Cubes and Squares. In: Aronov B., Basu S., Pach J., Sharir M. (eds) Discrete and Computational Geometry. Algorithms and Combinatorics, vol 25. Springer, Berlin, Heidelberg (2003). arXiv:math/0201108 [math.CO], 2002. See p. 3.
Fuzhen Zhang and Xiao-Dong Zhang, Comparison of the upper bounds for the extreme points of the polytopes of line-stochastic tensors, arXiv:2110.12337 [math.CO], 2021. See p. 3.
FORMULA
a(n)/A349507(n) ~ n^(-n^2)*(exp(-n)*n^(n-1/2)*(1+12*n))^(2*n)*(Pi/72)^n.
MATHEMATICA
Table[Numerator[n!^(2n)/(n^n^2)], {n, 10}]
PROG
(PARI) a(n) = numerator(n!^(2*n)/n^n^2); \\ Michel Marcus, Nov 22 2021
CROSSREFS
KEYWORD
nonn,frac
AUTHOR
Stefano Spezia, Nov 20 2021
STATUS
approved