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A183767 1/32 the number of (n+1) X 5 binary arrays with equal numbers of 2 X 2 subblocks with sum mod two being 0 and 1.

%I A183767 #18 Mar 12 2025 08:21:52
%S A183767 12,280,7392,205920,5912192,173065984,5134924800,153876579840,
%T A183767 4646469273600,141154845511680,4309194677698560,132086184685977600,
%U A183767 4062564621910867968,125316146802859048960,3875293808717379141632
%N A183767 1/32 the number of (n+1) X 5 binary arrays with equal numbers of 2 X 2 subblocks with sum mod two being 0 and 1.
%H A183767 R. H. Hardin, <a href="/A183767/b183767.txt">Table of n, a(n) for n = 1..200</a>
%F A183767 Empirically a(n) = 2^(3*n)*Gamma(2*n+1/2)/(Gamma(n+1/2)*n!). - _Peter Luschny_, Sep 24 2018
%F A183767 Conjectures from _Peter Bala_, Mar 11 2025: (Start)
%F A183767 1) a(n) = 2^n * binomial(4*n, 2*n).
%F A183767 2) a(n) = (4/3) * [x^n] T(3*n, 1 + x), where T(n, x) denotes the n-th Chebyshev polynomial of the first kind. (End)
%e A183767 Some solutions for 3 X 5:
%e A183767 ..0..0..1..1..0....0..0..0..1..1....1..0..1..1..1....0..0..0..1..1
%e A183767 ..0..0..0..0..0....1..1..1..1..1....1..1..0..0..1....0..1..1..1..0
%e A183767 ..0..0..0..1..0....0..1..0..0..1....1..1..1..1..1....0..0..0..0..1
%Y A183767 Column 4 of A183772.
%K A183767 nonn
%O A183767 1,1
%A A183767 _R. H. Hardin_, Jan 06 2011