A349508 - OEIS

%I #13 Dec 05 2021 10:40:48

%S 1,21318,111399602430962720,219754881677312748254868619396977023490,

%T 91574665590547903212939476569574243557076290573519342040406738188187312

%N a(n) is the numerator of binomial(n^3 + 6*n^2 - 6*n + 2, n^3 - 1)/n^3.

%C a(n) is the numerator of an 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 Chang et al. and Zhang et al.).

%H Haixia Chang, Vehbi Emrah Paksoy and Fuzhen Zhang, <a href="https://doi.org/10.1215/20088752-3605195">Polytopes of Stochastic Tensors</a>, Ann. Funct. Anal. 7(3): 386-393 (August 2016). <a href="https://arxiv.org/abs/1608.03203">arXiv:1608.03203 [math.CO]</a>, 2016. See p. 6.

%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)/A349509(n) <= A349510(n) < A349511(n) < A349512(n) (see Corollary 7 in Zhang et al., 2021).

%F a(n)/A349509(n) ~ 2^(-4 + 6*n - 6*n^2)*3^(-7/2 + 6*n - 6*n^2)*e^(-75 + 233/n + 18*n + 6*n^2)*n^(-1 - 6*n + 6*n^2)/sqrt(Pi).

%t a[n_]:=Numerator[Binomial[n^3+6n^2-6n+2,n^3-1]/n^3]; Array[a,6]

%Y Cf. A000578, A068601.

%Y Cf. A349506, A349507, A349509 (denominators), A349510, A349511, A349512.

%K nonn,frac

%O 1,2

%A _Stefano Spezia_, Nov 20 2021