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A290134 Number of unique X-rays of n X n binary matrices with exactly floor(n^2/2) ones.

%I A290134 #29 Mar 31 2018 16:58:46
%S A290134 1,1,2,5,14,42,130,415,1368,4603,15788,54863,193112,686049,2459942,
%T A290134 8881931,32292148,118038070,433790834,1601042055,5934546466,
%U A290134 22074679425,82399006636,308471888767,1158175006638,4359154749776,16447468190380,62188658733901
%N A290134 Number of unique X-rays of n X n binary matrices with exactly floor(n^2/2) ones.
%C A290134 The X-ray of a matrix is defined as the sequence of antidiagonal sums.
%C A290134 A unique X-ray allows reconstruction of the binary matrix.
%C A290134 The number of unique X-rays of all n X n binary matrices is A081294(n).
%C A290134 The number of all X-rays of n X n binary matrices is A010790(n).
%H A290134 Alois P. Heinz, <a href="/A290134/b290134.txt">Table of n, a(n) for n = 0..750</a>
%H A290134 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 A290134 <a href="/index/Mat#binmat">Index entries for sequences related to binary matrices</a>
%F A290134 a(n) ~ sqrt(3) * 2^(2*n-1) / (sqrt(Pi) * n^(3/2)). - _Vaclav Kotesovec_, Jul 22 2017
%e A290134 a(3) = 5: 00301, 02020, 10021, 10300, 12001.
%e A290134 a(4) = 14: 0004301, 0030320, 0034001, 0200321, 0204020, 0230021, 0230300, 1004021, 1004300, 1030301, 1034000, 1200320, 1204001, 1230020.
%p A290134 b:= proc(n, i, t) option remember; (m-> `if`(n>m, 0, `if`(n=m, 1,
%p A290134       b(n, i-t, 1-t)+`if`(i>n, 0, b(n-i, i-t, 1-t)))))(i*(i+1-t))
%p A290134     end:
%p A290134 a:= n-> b(iquo(n^2, 2), n, 1):
%p A290134 seq(a(n), n=0..40);
%t A290134 b[n_, i_, t_] := b[n, i, t] = Function[m, If[n > m, 0, If[n == m, 1, b[n, i-t, 1-t] + If[i > n, 0, b[n - i, i - t, 1 - t]]]]][i*(i + 1 - t)];
%t A290134 a[n_] := b[Quotient[n^2, 2], n, 1];
%t A290134 Table[a[n], {n, 0, 40}] (* _Jean-François Alcover_, Nov 06 2017, after _Alois P. Heinz_ *)
%Y A290134 Cf. A010790, A081294, A290133, A290058.
%K A290134 nonn
%O A290134 0,3
%A A290134 _Alois P. Heinz_, Jul 20 2017