A349510 a(n) = binomial(n^3-floor(((n-1)^3+1)/2), 3*n^2-3*n+1) + binomial(n^3-floor(((n-1)^3+2)/2), 3*n^2-3*n+1).

0, 1, 2, 10395, 709721037200, 11641222531417506431654250, 94310884171276301089942905465465961965897600, 1948497841630989653689709780233830548909045113177792777217829860522656, 192558458967017735390472923791964989275151544601992192306693834632003663346431678074519409150869009600

Offset

0,3

Comments

a(n) is a sharp 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 Li et al. and Zhang et al.).

Links

Table of n, a(n) for n=0..8.

Zhongshan Li, Fuzhen Zhang and Xiao-Dong Zhang, On the number of vertices of the stochastic tensor polytope, Linear and Multilinear Algebra, 65:10, 2064-2075, (2017). arXiv:1702.04288 [math.CO], 2017. See p. 4.

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. 4.

Formula

A349508(n)/A349509(n) <= a(n) < A349511(n) < A349512(n) (see Corollary 7 in Zhang et al., 2021).

a(n) ~ (n/6)^(3*n*(n-1))*exp(-6+13/n+3*n^2)/(3*sqrt(6*Pi)).

Mathematica

a[n_]:=Binomial[n^3-Floor[((n-1)^3+1)/2], 3n^2-3n+1]+Binomial[n^3-Floor[((n-1)^3+2)/2], 3n^2-3n+1]; Array[a, 9, 0]

Crossrefs

Cf. A242658.

Cf. A349506, A349507, A349508, A349509, A349511, A349512.

Sequence in context: A086563 A023328 A294324 * A134656 A128122 A082178

Adjacent sequences: A349507 A349508 A349509 * A349511 A349512 A349513

Keyword

nonn

Author

Stefano Spezia, Nov 20 2021

Status

approved