A349511 -id:A349511 - cp's OEIS frontend

A349509 a(n) is the denominator of binomial(n^3 + 6n^2 - 6n + 2, n^3 - 1)/n^3.

1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 3, 1, 1, 1, 1, 1, 1

Offset: 1

Stefano Spezia, Nov 20 2021

a(n) is the denominator 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.).
Conjecture: 1 and 3 are the only terms that appear in this sequence.
This conjecture is correct, see formula. - Kevin Ryde, Jul 01 2023

Kevin Ryde, Table of n, a(n) for n = 1..10000
Haixia Chang, Vehbi Emrah Paksoy, and Fuzhen Zhang, Polytopes of Stochastic Tensors, Ann. Funct. Anal. 7(3): 386-393 (August 2016). arXiv:1608.03203 [math.CO], 2016. See p. 6.
Kevin Ryde, PARI/GP Code and Notes
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.

Cf. A000578, A068601.
Cf. A349506, A349507, A349508 (numerators), A349510, A349511, A349512.
Cf. A363739 (run lengths), A349929 (indices of 3's).

a[n_]:=Denominator[Binomial[n^3+6n^2-6n+2,n^3-1]/n^3]; Array[a,90]

PARI
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from math import gcd, comb
def A349509(n): return n**3//gcd(comb(n*(n*(n + 6) - 6) + 2,6*n*(n-1)+3),n**3) # Chai Wah Wu, Dec 06 2021

A349508(n)/a(n) <= A349510(n) < A349511(n) < A349512(n) (see Corollary 7 in Zhang et al., 2021).
A349508(n)/a(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).
a(n) = 1 if n=1 or any x[i] + y[i] >= 3 where x and y are the ternary digits of n^3 = Sum x[i]*3^i and 6*n^2 - 6*n + 3 = Sum y[i]*3^i; and a(n) = 3 otherwise. - Kevin Ryde, Jul 01 2023

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

Original entry on oeis.org

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

Offset: 1

Stefano Spezia, Nov 20 2021

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.).

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.

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

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

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

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

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

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

Original entry on oeis.org

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

Offset: 1

Stefano Spezia, Nov 20 2021

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.).

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.

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

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

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

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

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

A349508 a(n) is the numerator of binomial(n^3 + 6n^2 - 6n + 2, n^3 - 1)/n^3.

Original entry on oeis.org

1, 21318, 111399602430962720, 219754881677312748254868619396977023490, 91574665590547903212939476569574243557076290573519342040406738188187312

Offset: 1

Stefano Spezia, Nov 20 2021

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.).

Haixia Chang, Vehbi Emrah Paksoy and Fuzhen Zhang, Polytopes of Stochastic Tensors, Ann. Funct. Anal. 7(3): 386-393 (August 2016). arXiv:1608.03203 [math.CO], 2016. See p. 6.
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.

Cf. A000578, A068601.

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

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

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

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).

A349510 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).

Original entry on oeis.org

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

Offset: 0

Stefano Spezia, Nov 20 2021

nonn

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.).

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.

Cf. A242658.

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

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]

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)).

A349512 a(n) = binomial(n^3 + 3n^2 - 3n + 1, n^3).

Original entry on oeis.org

1, 2, 6435, 4154246671960, 5397234129638871133346507775, 80240648651400365471854502514501453704175376562496, 54198670627270688013781273396239242514947489935351300645194042280183395324517200

Offset: 0

Stefano Spezia, Nov 20 2021

nonn

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 Zhang et al.).

Fuzhen Zhang and Xiao-Dong Zhang, Enumerating extreme points of the polytopes of stochastic tensors: an optimization approach, Optimization, 69:4, 729-741, (2020). arXiv:2008.04655 [math.CO], 2020. See p. 6.
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. 5.

Cf. A000578, A229013.

Cf. A349506, A349507, A349508, A349509, A349510, A349511.

a[n_]:=Binomial[n^3+3n^2-3n+1,n^3]; Array[a,8,0]

A349508(n)/A349509(n) <= A349510(n) < A349511(n) < a(n) (see Corollary 7 in Zhang et al., 2021).

a(n) ~ C*3^(3(n - n^2))*exp(3*(3*n/2 + n^2))*n^(3(-n + n^2)), where C = e^(-15)/sqrt(54*Pi).

cp's OEIS Frontend

A349509 a(n) is the denominator of binomial(n^3 + 6*n^2 - 6*n + 2, n^3 - 1)/n^3.

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

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

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Programs

Mathematica

PARI

Formula

A349508 a(n) is the numerator of binomial(n^3 + 6*n^2 - 6*n + 2, n^3 - 1)/n^3.

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Mathematica

Formula

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).

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Formula

A349512 a(n) = binomial(n^3 + 3*n^2 - 3*n + 1, n^3).

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Programs

Mathematica

Formula

A349509 a(n) is the denominator of binomial(n^3 + 6n^2 - 6n + 2, n^3 - 1)/n^3.

A349508 a(n) is the numerator of binomial(n^3 + 6n^2 - 6n + 2, n^3 - 1)/n^3.

A349510 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).

A349512 a(n) = binomial(n^3 + 3n^2 - 3n + 1, n^3).