|
%I #19 Mar 04 2019 23:08:13
%S 1,4,34,500,10900,322768,12297768,580849872,33093252880,2227152575552,
%T 174131286983712,15604440074084672,1584856558077903168,
%U 180712593036822482176,22946861101272125055616,3222156375409363475703040
%N 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).
%C From _Gus Wiseman_, Mar 01 2019: (Start)
%C 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:
%C o-o-o- -o
%C | | | ... |
%C o-o-o- -o
%C (End)
%H R. H. Hardin, <a href="/A207864/b207864.txt">Table of n, a(n) for n = 1..61</a>
%F 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
%F 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
%e Some solutions for n=5:
%e 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1
%e 1 0 1 0 1 2 1 2 1 0 1 0 1 2 1 0 1 0 1 0
%e 0 1 0 1 0 1 0 1 2 1 0 1 0 1 0 2 2 1 0 1
%e 1 2 1 0 1 0 1 3 3 0 2 0 3 2 2 1 1 0 1 2
%e 0 1 0 1 2 1 2 4 1 2 0 1 0 1 0 2 0 1 2 0
%t Table[Expand[x*(x-1)*(x^2-3*x+3)^(n-1)]/.x^k_.->BellB[k],{n,20}] (* _Gus Wiseman_, Mar 01 2019 *)
%Y Column 2 of A207868.
%Y Cf. A000110, A000569, A109808, A229048, A240936, A321750, A321979, A321980, A321982, A322064.
%K nonn
%O 1,2
%A _R. H. Hardin_, Feb 21 2012
| 