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%I #7 Jun 24 2014 01:08:34
%S 1,1,1,1,2,1,1,5,3,1,1,8,12,4,1,1,13,28,22,5,1,1,18,63,68,35,6,1,1,25,
%T 117,197,135,51,7,1,1,32,208,464,480,236,70,8,1,1,41,336,1016,1376,
%U 996,378,92,9,1,1,50,525,2000,3600,3372,1848,568,117,10,1,1,61,775,3725
%N Triangle read by rows giving T(n,k) = number of sets of k black squares in an n X n checkerboard with the upper left corner colored black, such that the line joining any 2 squares slopes down to the right, 0 <= k <= n.
%C Based on a question from Cees H. Elzinga (ch.elzinga(AT)tiscali.nl), Dec 30 2002
%F More generally, let f(m, n, k) be the number of such sets in an m X n checkerboard. Then f(m, n, k) = sum_{k-1<=m'<m, k-1<=n'<n, m+n+m'+n' even} f(m', n', k-1).
%F G.f.: sum_{m>=0, n>=0, k>=0} f(m, n, k) x^m y^n z^k = (1+x)(1+y)/((1-x^2)(1-y^2)+x y z(1+x y)).
%F T(n, 0) = T(n, n) = 1. T(n, 1) = ceiling(n^2/2). T(n, 2) = (n^2 (n^2-2n+4))/16 if n is even, ((n-1)^2 (n^2+3))/16 if n is odd. T(n, n-1) = n. T(n, n-2) = (n-1)(3n-4)/2.
%F G.f. (conjectured): sum_{n>=0, k>=0} T(n, k) x^n y^k = sqrt((1+x)/((1+x-x y)((1-x)^2 - x y(1+x)))).
%F Conjecture: sum_{k=0..n} T(n, k) = A025565(n+1).
%e T(5,3)=22; one of the 22 sets of 3 is shown by the asterisks below; the 'o's denote black squares not in the set.
%e *.o.o
%e .*.o.
%e o.o.*
%e .o.o.
%e o.o.o
%t f[m_, n_, 0] := 1; f[m_, n_, k_] := f[m, n, k]=Sum[If[EvenQ[m+n+mp+np], f[mp, np, k-1], 0], {mp, k-1, m-1}, {np, k-1, n-1}]; T[n_, k_] := f[n, n, k]; Flatten[Table[T[n, k], {n, 0, 11}, {k, 0, n}]]
%Y Cf. A025565.
%K nonn,easy,tabl
%O 0,5
%A _Dean Hickerson_, Jan 02 2003
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