A277443 - OEIS

%I #14 Sep 21 2019 20:35:52

%S 0,0,0,0,2,0,0,0,18,0,0,2,12,84,0,0,0,114,264,260,0,0,2,180,2652,1920,

%T 630,0,0,0,858,16080,29660,8520,1302,0,0,2,1932,119844,367080,198030,

%U 28140,2408,0,0,0,7074,816984,4843460,4067280,932862,76272,4104,0,0,2,18660,5784492,62682480,85847910,28576380,3440024,179424,6570,0

%N Square array A(n,k) (n>=1, k>=1) read by antidiagonals: A(n,k) is the number of n-colorings of the prism graph Y_k on 2k vertices.

%C Y_1 contains a loop, so has no colorings with any number of colors. Y_2 is the cycle graph C_4 with two double edges; these two graphs are therefore equivalent with respect to number of colorings.

%H N. L. Biggs, R. M. Damerell and D. A. Sands, <a href="https://dx.doi.org/10.1016%2F0095-8956%2872%2990016-0">Recursive families of graphs</a>, Journal of Combinatorial Theory Series B Volume 12 (1972), 123-131. MR0294172

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PrismGraph.html">Prism Graph</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Chromatic_polynomial">Chromatic polynomial</a>

%F A(n,k) = (n^2-3n+3)^k+(n-1)((3-n)^k+(1-n)^k)+n^2-3n+1.

%e Square array A(n,k) begins:

%e   0,   0,    0,      0,       0,        0,          0, ...

%e   0,   2,    0,      2,       0,        2,          0, ...

%e   0,  18,   12,    114,     180,      858,       1932, ...

%e   0,  84,  264,   2652,   16080,   119844,     816984, ...

%e   0, 260, 1920,  29660,  367080,  4843460,   62682480, ...

%e   0, 630, 8520, 198030, 4067280, 85847910, 1800687000, ...

%Y Cf. A277444 (colorings of Möbius ladders), A182406 (square grid graphs).

%Y Columns k=1,2 are A000004 and A091940.

%Y Rows n=1,2 are A000004 and A010673.

%K nonn,tabl

%O 1,5

%A _Jeremy Tan_, Oct 15 2016