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%I #27 Feb 19 2019 20:17:15
%S 1,1,1,1,3,4,3,1,1,5,13,23,30,30,23,13,5,1,1,7,26,68,139,234,334,411,
%T 440,411,334,234,139,68,26,7,1,1,9,43,145,386,860,1660,2838,4362,6090,
%U 7779,9135,9892,9892,9135,7779,6090,4362,2838,1660,860,386,145,43,9,1
%N Number T(n,k) of X-rays of n X n binary matrices with exactly k ones; triangle T(n,k), n>=0, 0<=k<=n^2, read by rows.
%C The X-ray of a matrix is defined as the sequence of antidiagonal sums.
%C T(n,k) is defined for all n,k >= 0. The triangle contains only the positive terms. T(n,k) = 0 for k>n^2.
%H Alois P. Heinz, <a href="/A290057/b290057.txt">Rows n = 0..40, flattened</a>
%H C. Bebeacua, T. Mansour, A. Postnikov and S. Severini, <a href="https://arxiv.org/abs/math/0506334">On the X-rays of permutations</a>, arXiv:math/0506334 [math.CO], 2005.
%H <a href="/index/Mat#binmat">Index entries for sequences related to binary matrices</a>
%F T(n,floor(n^2/2)) = A290058(n).
%F T(n,k) = T(n,n^2-k).
%e Triangle T(n,k) begins:
%e 1;
%e 1, 1;
%e 1, 3, 4, 3, 1;
%e 1, 5, 13, 23, 30, 30, 23, 13, 5, 1;
%e 1, 7, 26, 68, 139, 234, 334, 411, 440, 411, 334, 234, 139, 68, 26, 7, 1;
%e ...
%p b:= proc(n, i, t) option remember; (m-> `if`(n>m, 0, `if`(n=m, 1,
%p add(b(n-j, i-t, 1-t), j=0..min(i, n)))))(i*(i+1-t))
%p end:
%p T:= (n, k)-> b(k, n, 1):
%p seq(seq(T(n, k), k=0..n^2), n=0..7);
%t b[n_,i_,t_]:= b[n, i, t] = Function[{m, jm}, If[n>m, 0, If[n==m, 1, Sum[b[n-j, i-t, 1-t], {j, 0, jm}]]]][i*(i+1-t), Min[i, n]]; T[n_, k_]:= b[k, n, 1]; Table[T[n, k], {n, 0, 7}, {k, 0, n^2}] // Flatten (* _Jean-François Alcover_, Aug 09 2017, translated from Maple *)
%Y Columns k=0-2 give: A000012, A004273, A091823(n-1) for n>1.
%Y Main diagonal gives A290052.
%Y Row sums give A010790.
%Y Cf. A000290, A290058.
%K nonn,tabf
%O 0,5
%A _Alois P. Heinz_, Jul 19 2017
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