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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). 3
1, 4, 34, 500, 10900, 322768, 12297768, 580849872, 33093252880, 2227152575552, 174131286983712, 15604440074084672, 1584856558077903168, 180712593036822482176, 22946861101272125055616, 3222156375409363475703040 (list; graph; refs; listen; history; text; internal format)
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

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Last modified November 29 18:41 EST 2021. Contains 349416 sequences. (Running on oeis4.)