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Number of X-rays of n X n binary matrices with exactly floor(n^2/2) ones.
3

%I #20 Aug 09 2017 16:26:30

%S 1,1,4,30,440,9892,331950,15121926,915201732,70120569074,

%T 6696969703276,774618119733020,107284227278413622,

%U 17455779156567652806,3307802489634916900474,720231707823173636419042,178973636259170839478327332,50249140887232774758578932120

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

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

%H Alois P. Heinz, <a href="/A290058/b290058.txt">Table of n, a(n) for n = 0..253</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) = A290057(n,floor(n^2/2)).

%F a(n) ~ 6*sqrt(Pi) * n^(2*n+1/2) / exp(2*n). - _Vaclav Kotesovec_, Jul 22 2017

%e a(2) = 4: 011, 020, 101, 110.

%e a(3) = 30: 00121, 00211, 00220, 00301, 00310, 01021, 01111, 01120, 01201, 01210, 01300, 02011, 02020, 02101, 02110, 02200, 10021, 10111, 10120, 10201, 10210, 10300, 11011, 11020, 11101, 11110, 11200, 12001, 12010, 12100.

%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 a:= n-> b(iquo(n^2, 2), n, 1):

%p seq(a(n), n=0..20);

%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]]; a[n_]:= b[Quotient[n^2, 2], n, 1]; Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Aug 09 2017, translated from Maple *)

%Y Cf. A290057, A290134.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Jul 19 2017