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

1,3

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.

Links

Table of n, a(n) for n=1..66.

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

Formula

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

Example

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, ...

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.

Sequence in context: A098643 A193474 A241020 * A274710 A028625 A344441

Adjacent sequences: A277441 A277442 A277443 * A277445 A277446 A277447

Keyword

nonn,tabl

Author

Jeremy Tan, Oct 15 2016

Status

approved