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A207864
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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).
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4
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1, 4, 34, 500, 10900, 322768, 12297768, 580849872, 33093252880, 2227152575552, 174131286983712, 15604440074084672, 1584856558077903168, 180712593036822482176, 22946861101272125055616, 3222156375409363475703040
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OFFSET
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1,2
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COMMENTS
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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)
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LINKS
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FORMULA
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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
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EXAMPLE
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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
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MATHEMATICA
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Table[Expand[x*(x-1)*(x^2-3*x+3)^(n-1)]/.x^k_.->BellB[k], {n, 20}] (* Gus Wiseman, Mar 01 2019 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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