A349506 a(n) is the numerator of n!^(2*n)/(n^n^2).

1, 1, 64, 6561, 63403380965376, 1000000000000, 10061319724179153710638694400000000000000, 9396559338406702410023114843902587890625, 528450425551613768181656289451784661530463698944000000000000000000, 13597557929083423616920569866317288159544321459878738801559053666747416576

Offset: 1

Author

Stefano Spezia, Nov 20 2021

Keywords

• nonn
• frac

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.

Crossrefs

Cf. A000142, A000290, A002489, A005843, A185141.

Cf. A349507 (denominators), A349508, A349509, A349510, A349511, A349512.

Programs

Mathematica

Table[Numerator[n!^(2n)/(n^n^2)],{n,10}]

PARI

a(n) = numerator(n!^(2*n)/n^n^2); \\ Michel Marcus, Nov 22 2021

Formula

a(n)/A349507(n) ~ n^(-n^2)*(exp(-n)*n^(n-1/2)*(1+12*n))^(2*n)*(Pi/72)^n.