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A290057 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.

%I A290057 #27 Feb 19 2019 20:17:15
%S A290057 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 A290057 440,411,334,234,139,68,26,7,1,1,9,43,145,386,860,1660,2838,4362,6090,
%U A290057 7779,9135,9892,9892,9135,7779,6090,4362,2838,1660,860,386,145,43,9,1
%N A290057 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 A290057 The X-ray of a matrix is defined as the sequence of antidiagonal sums.
%C A290057 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 A290057 Alois P. Heinz, <a href="/A290057/b290057.txt">Rows n = 0..40, flattened</a>
%H A290057 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 A290057 <a href="/index/Mat#binmat">Index entries for sequences related to binary matrices</a>
%F A290057 T(n,floor(n^2/2)) = A290058(n).
%F A290057 T(n,k) = T(n,n^2-k).
%e A290057 Triangle T(n,k) begins:
%e A290057   1;
%e A290057   1, 1;
%e A290057   1, 3,  4,  3,   1;
%e A290057   1, 5, 13, 23,  30,  30,  23,  13,   5,   1;
%e A290057   1, 7, 26, 68, 139, 234, 334, 411, 440, 411, 334, 234, 139, 68, 26, 7, 1;
%e A290057   ...
%p A290057 b:= proc(n, i, t) option remember; (m-> `if`(n>m, 0, `if`(n=m, 1,
%p A290057       add(b(n-j, i-t, 1-t), j=0..min(i, n)))))(i*(i+1-t))
%p A290057     end:
%p A290057 T:= (n, k)-> b(k, n, 1):
%p A290057 seq(seq(T(n, k), k=0..n^2), n=0..7);
%t A290057 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 A290057 Columns k=0-2 give: A000012, A004273, A091823(n-1) for n>1.
%Y A290057 Main diagonal gives A290052.
%Y A290057 Row sums give A010790.
%Y A290057 Cf. A000290, A290058.
%K A290057 nonn,tabf
%O A290057 0,5
%A A290057 _Alois P. Heinz_, Jul 19 2017