Marc Chamberland, <a href="https://doi.org/10.1016/j.laa.2011.08.030">Factored matrices can generate combinatorial identities</a>,Linear Algebra and its ApplicationsVolume 438, Issue 4, 15 February 2013, Pages 1667-1677,.

Marc Chamberland, <a href="https://doi.org/10.1016/j.laa.2011.08.030">Factored matrices can generate combinatorial identities</a>,Linear Algebra and its ApplicationsVolume 438, Issue 4, 15 February 2013, Pages 1667-1677,

LU decomposition of square array = A051731 * transpose(A054522) (see Chamberland, p. 1673). - Peter Bala, Oct 15 2023

Cf. A001088, A003990, A003991, A050873, A051731, A054431, A001088A054522.

approved

editing

editing

approved

x\y 1 2 3 4 5 6 ...

[1, 1, 1, 1, 1, 1, 1, 1, 1, 1: , 1 , 1 , 1 , 1 , 1 , 1 ...]

[1, 2, 1, 2, 1, 2, 1, 2, 1, 2: , 1 , 2 , 1 , 2 , 1 , 2 ...]

[1, 1, 3, 1, 1, 3, 1, 1, 3: , 1 , 1 , 3 , 1 , 1 , 3 ..., 1]

[1, 2, 1, 4, 1, 2, 1, 4: , 1 , 2 , 1 , 4 , 1 , 2 ..., 1, 4]

[1, 1, 1, 1, 5, 1, 1, 1, 1, 5: , 1 , 1 , 1 , 1 , 5 , 1 ...]

[1, 2, 3, 2, 1, 6, 1, 2, 3, 2, 1, 6, 1, 2, 3, 2]

[1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 7, 1, 1]

[1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 2, 1, 8]

[1, 1, 3, 1, 1, 3, 1, 1, 9, 1, 1, 3, 1, 1, 3, 1]

[1, 2, 1, 2, 5, 2, 1, 2, 1, 10, 1, 2, 1, 2, 5, 2]

[1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 1, 1, 1, 1, 1]

[1, 2, 3, 4, 1, 6, 1, 4, 3, 2, 1, 12, 1, 2, 3, 4]

approved

editing

proposed

approved

editing

proposed

Grant Cairns, <a href="httphttps://www.combinatoricsdoi.org/Volume_8/Abstracts/v8i1n610.html37236/1591">Queens on Non-square Tori</a>, El. J. Combinatorics, N6, 2001.

proposed

editing

editing

proposed

Dirichlet generating function: Sum_{ n>=1 } Sum_{ k>=1 } gcd(n, k)/n^s/k^c = zeta(s)*zeta(c)*zeta(s + c - 1)/zeta(s + c). - Mats Granvik, Feb 13 2021

Dirichlet generating function: Sum_{ n>=1 } Sum_{ k>=1 } gcd(n, k)/n^s/k^c = zeta(s)*zeta(c)*zeta(s + c - 1)/zeta(s + c). - Mats Granvik, Feb 13 2021