cp's OEIS Frontend

A212209 Square array A(n,k), n>=1, k>=1, read by antidiagonals: A(n,k) is the number of n-colorings of the square diagonal grid graph DG_(k,k).

%I A212209 #20 Feb 16 2025 08:33:17
%S A212209 1,0,2,0,0,3,0,0,0,4,0,0,0,24,5,0,0,0,72,120,6,0,0,0,168,6720,360,7,0,
%T A212209 0,0,360,935040,126360,840,8,0,0,0,744,325061760,265035240,1128960,
%U A212209 1680,9,0,0,0,1512,283192323840,3322711053720,17160407040,6510000,3024,10
%N A212209 Square array A(n,k), n>=1, k>=1, read by antidiagonals: A(n,k) is the number of n-colorings of the square diagonal grid graph DG_(k,k).
%C A212209 The square diagonal grid graph DG_(n,n) has n^2 = A000290(n) vertices and 2*(n-1)*(2*n-1) = A002943(n-1) edges; see A212208 for example. The chromatic polynomial of DG_(n,n) has n^2+1 = A002522(n) coefficients.
%C A212209 This graph is also called the king graph. - _Andrew Howroyd_, Jun 25 2017
%H A212209 Andrew Howroyd, <a href="/A212209/b212209.txt">Table of n, a(n) for n = 1..153</a>
%H A212209 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/KingGraph.html">King Graph</a>
%H A212209 Wikipedia, <a href="https://en.wikipedia.org/wiki/Chromatic_polynomial">Chromatic Polynomial</a>
%e A212209 Square array A(n,k) begins:
%e A212209   1,   0,       0,           0,                0, ...
%e A212209   2,   0,       0,           0,                0, ...
%e A212209   3,   0,       0,           0,                0, ...
%e A212209   4,  24,      72,         168,              360, ...
%e A212209   5, 120,    6720,      935040,        325061760, ...
%e A212209   6, 360,  126360,   265035240,    3322711053720, ...
%e A212209   7, 840, 1128960, 17160407040, 2949948395735040, ...
%Y A212209 Columns 1-5 give: A000027, A052762 = 24*A000332, 24*A068250, 24*A068251, 24*A068252.
%Y A212209 Rows n=1-16 give: A000007, A000038, 3*A000007, 4*A068293, 5*A068294, 6*A068295, 7*A068296, 8*A068297, 9*A068298, 10*A068299, 11*A068300, 12*A068301, 13*A068302, 14*A068303, 15*A068304, 16*A068305.
%Y A212209 Cf. A000290, A002943, A212208, A208021.
%K A212209 nonn,tabl
%O A212209 1,3
%A A212209 _Alois P. Heinz_, May 04 2012