A207864 Number of n X 2 nonnegative integer arrays with new values 0 upwards introduced in row major order and no element equal to any horizontal or vertical neighbor (colorings ignoring permutations of colors).

1, 4, 34, 500, 10900, 322768, 12297768, 580849872, 33093252880, 2227152575552, 174131286983712, 15604440074084672, 1584856558077903168, 180712593036822482176, 22946861101272125055616, 3222156375409363475703040

Offset

1,2

Comments

From Gus Wiseman, Mar 01 2019: (Start)

Also the number of stable partitions of the n-ladder graph. A stable partition of a graph is a set partition of the vertices where no edge has both ends in the same block. The n-ladder has 2n vertices and looks like:

o-o-o- -o

| | | ... |

o-o-o- -o

(End)

Links

R. H. Hardin, Table of n, a(n) for n = 1..61

Formula

It appears that the sequence terms are given by the Dobinski-type formula a(n+1) = (1/e) * Sum_{k>=0} (1+k+k^2)^n/k!. - Peter Bala, Mar 12 2012

Apply x^n -> B(n) to the polynomial chi(n) = x (x - 1) (x^2 - 3 x + 3)^(n - 1), where B = A000110. - Gus Wiseman, Mar 01 2019

Example

Some solutions for n=5:
  0 1   0 1   0 1   0 1   0 1   0 1   0 1   0 1   0 1   0 1
  1 0   1 0   1 2   1 2   1 0   1 0   1 2   1 0   1 0   1 0
  0 1   0 1   0 1   0 1   2 1   0 1   0 1   0 2   2 1   0 1
  1 2   1 0   1 0   1 3   3 0   2 0   3 2   2 1   1 0   1 2
  0 1   0 1   2 1   2 4   1 2   0 1   0 1   0 2   0 1   2 0

Mathematica

Table[Expand[x*(x-1)*(x^2-3*x+3)^(n-1)]/.x^k_.->BellB[k], {n, 20}] (* Gus Wiseman, Mar 01 2019 *)

Crossrefs

Column 2 of A207868.

Cf. A000110, A000569, A109808, A229048, A240936, A321750, A321979, A321980, A321982, A322064.

Sequence in context: A002105 A198717 A198908 * A222077 A081972 A158961

Adjacent sequences: A207861 A207862 A207863 * A207865 A207866 A207867

Keyword

nonn

Author

R. H. Hardin, Feb 21 2012

Status

approved