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).

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, 0, 0, 360, 935040, 126360, 840, 8, 0, 0, 0, 744, 325061760, 265035240, 1128960, 1680, 9, 0, 0, 0, 1512, 283192323840, 3322711053720, 17160407040, 6510000, 3024, 10

Offset

1,3

Comments

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.

This graph is also called the king graph. - Andrew Howroyd, Jun 25 2017

Links

Andrew Howroyd, Table of n, a(n) for n = 1..153

Eric Weisstein's World of Mathematics, King Graph

Wikipedia, Chromatic Polynomial

Example

Square array A(n,k) begins:
  1,   0,       0,           0,                0, ...
  2,   0,       0,           0,                0, ...
  3,   0,       0,           0,                0, ...
  4,  24,      72,         168,              360, ...
  5, 120,    6720,      935040,        325061760, ...
  6, 360,  126360,   265035240,    3322711053720, ...
  7, 840, 1128960, 17160407040, 2949948395735040, ...

Crossrefs

Columns 1-5 give: A000027, A052762 = 24*A000332, 24*A068250, 24*A068251, 24*A068252.

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.

Cf. A000290, A002943, A212208, A208021.

Sequence in context: A140579 A132681 A127648 * A259481 A132825 A259480

Adjacent sequences: A212206 A212207 A212208 * A212210 A212211 A212212

Keyword

nonn,tabl

Author

Alois P. Heinz, May 04 2012

Status

approved