%I #25 May 06 2018 11:59:43

%S 1,1,2,5,8,13,21,31,45,65,92,127,175,237,318,425,561,735,959,1241,

%T 1597,2047,2607,3305,4174,5247,6569,8197,10189,12621,15588,19189,

%U 23551,28829,35189,42841,52033,63039,76197,91903,110603,132831,159215,190463,227416

%N Number of unique X-rays of n X n binary matrices with exactly n ones.

%C The X-ray of a matrix is defined as the sequence of antidiagonal sums.

%C A unique X-ray allows reconstruction of the binary matrix.

%H Alois P. Heinz, <a href="/A290133/b290133.txt">Table of n, a(n) for n = 0..10000</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 a(n) ~ exp(Pi*sqrt(2*n/3)) / (2^(9/4) * 3^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, May 06 2018

%e a(3) = 5: 00021, 00300, 02001, 10020, 12000.

%e a(4) = 8: 0000301, 0004000, 0030001, 0200020, 1000021, 1000300, 1030000, 1200001.

%p b:= proc(n, i) option remember; (m-> `if`(n>m, 0,

%p `if`(n=m or n=0, 1, add(b(n-i*j, min(n-i*j, i-1))*

%p `if`(j=1, 2, 1), j=0..min(2, n/i)))))(i*(i+1))

%p end:

%p a:= n-> `if`(n=0, 1, 1+b(n, n-1)) :

%p seq(a(n), n=0..60);

%t b[n_, i_] := b[n, i] = Function [m, If[n > m, 0, If[n == m || n == 0, 1, Sum[b[n - i*j, Min[n - i*j, i - 1]]*If[j == 1, 2, 1], {j, 0, Min[2, n/i]} ]]]][i*(i + 1)];

%t a[n_] := If[n == 0, 1, 1 + b[n, n - 1]] ;

%t Table[a[n], {n, 0, 60}] (* _Jean-François Alcover_, Nov 07 2017, after _Alois P. Heinz_ *)

%Y Cf. A290052, A290134.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Jul 20 2017