A349507 a(n) is the denominator of n!^(2*n)/(n^n^2).

1, 1, 27, 256, 30517578125, 531441, 378818692265664781682717625943, 1208925819614629174706176, 8727963568087712425891397479476727340041449, 867361737988403547205962240695953369140625, 12527829399838427440107579247354215251149392000034969484678615956504532008683916069945559954314411495091

Offset

1,3

Comments

a(n) is the denominator 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..28

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

A349506(n)/a(n) ~ n^(-n^2)*(exp(-n)*n^(n-1/2)*(1+12*n))^(2*n)*(Pi/72)^n.

Mathematica

Table[Denominator[n!^(2n)/(n^n^2)], {n, 11}]

Prog

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

Crossrefs

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

Cf. A349506 (numerators), A349508, A349509, A349510, A349511, A349512.

Sequence in context: A016107 A101379 A132652 * A373039 A125390 A126548

Adjacent sequences: A349504 A349505 A349506 * A349508 A349509 A349510

Keyword

nonn,frac

Author

Stefano Spezia, Nov 20 2021

Status

approved