A277444 Square array A(n,k) (n>=1, k>=1) read by antidiagonals: A(n,k) is the number of n-colorings of the Möbius ladder M_k on 2k vertices.

0, 0, 2, 0, 0, 6, 0, 2, 0, 12, 0, 0, 42, 24, 20, 0, 2, 48, 420, 120, 30, 0, 0, 306, 2160, 2420, 360, 42, 0, 2, 600, 17532, 27600, 9750, 840, 56, 0, 0, 2442, 115464, 375260, 191760, 30702, 1680, 72, 0, 2, 6048, 830100, 4810680, 4098510, 917280, 81032, 3024, 90, 0, 0, 20706, 5745120, 62813540, 85691640, 28669662, 3406368, 187560, 5040, 110

Offset: 1

Author

Jeremy Tan, Oct 15 2016

Keywords

• nonn
• tabl

Comments

M_1 is two vertices connected by a triple edge and thus behaves like the path graph P_2 in terms of colorings. M_2 is isomorphic to K_4, the tetrahedral graph.

Examples

			Square array A(n,k) begins:
0,    0,    0,      0,       0,        0,          0, ...
2,    0,    2,      0,       2,        0,          2, ...
6,    0,   42,     48,     306,      600,       2442, ...
12,  24,  420,   2160,   17532,   115464,     830100, ...
20, 120, 2420,  27600,  375260,  4810680,   62813540, ...
30, 360, 9750, 191760, 4098510, 85691640, 1801468230, ...
		

Links

• N. L. Biggs, R. M. Damerell and D. A. Sands, Recursive families of graphs, Journal of Combinatorial Theory Series B Volume 12 (1972), 123-131. MR0294172
• Eric Weisstein's World of Mathematics, Möbius Ladder
• Wikipedia, Chromatic polynomial

Crossrefs

Cf. A277443 (colorings of prism graphs), A182406 (square grid graphs).

Columns k=1,2 are A002378 and A052762. Rows n=1,2 are A000004 and A010673.

Formula

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