cp's OEIS Frontend

A079213 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.

%I A079213 #9 Aug 04 2025 19:09:56
%S A079213 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 A079213 117,197,135,51,7,1,1,32,208,464,480,236,70,8,1,1,41,336,1016,1376,
%U A079213 996,378,92,9,1,1,50,525,2000,3600,3372,1848,568,117,10,1,1,61,775,3725
%N A079213 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 A079213 Based on a question from Cees H. Elzinga (ch.elzinga(AT)tiscali.nl), Dec 30 2002
%F A079213 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 A079213 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 A079213 T(n, 0) = T(n, n) = 1. T(n, 1) = ceiling(n^2/2). T(n, 2) = (n^2 * (n^2-2*n+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 A079213 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 A079213 Conjecture: Sum_{k=0..n} T(n, k) = A025565(n+1).
%e A079213 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 A079213 *.o.o
%e A079213 .*.o.
%e A079213 o.o.*
%e A079213 .o.o.
%e A079213 o.o.o
%t A079213 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 A079213 Cf. A025565.
%K A079213 nonn,easy,tabl
%O A079213 0,5
%A A079213 _Dean Hickerson_, Jan 02 2003