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%I #17 Aug 28 2022 08:23:08
%S 1,0,2,0,12,0,90,0,810,0,8940,0,116760,0,1756860,0,29933820,0,
%T 569744280,0,11981526480,0,275893362360,0,6903968231160,0,
%U 186558764792400,0,5413973807642400
%N Number of n X n arrays of squares of integers, symmetric under 90-degree rotation, with all rows summing to 2.
%C a(2*n) is the number of n X n 0-1 matrices A such that all row sums of A + A^T equal 2. - _Robert Israel_, Feb 19 2019
%H Robert Israel, <a href="/A156431/b156431.txt">Table of n, a(n) for n = 2..807</a>
%H Robert Israel, <a href="/A156431/a156431.pdf">Proof of recurrence</a>
%F From _Robert Israel_, Feb 18 2019: (Start)
%F a(2k+1) = 0.
%F a(2k) = 2*(k-1)*(k-2)*a(2k-6) - (k-1)*a(2k-4) + (2k-1)*a(2k-2).
%F (End)
%p f:= gfun:-rectoproc({b(k) = 2*(k-1)*(k-2)*b(k-3)-(k-1)*b(k-2)+(2*k-1)*b(k-1), b(0)=1,b(1)=1,b(2)=2},b(k),remember):
%p seq(op([f(k),0]),k=0..50); # _Robert Israel_, Feb 18 2019
%t a[n_] := a[n] = If[OddQ[n], 0, Switch[n, 2, 1, 4, 2, 6, 12, _, (n-4)*(n/2-1)*a[n-6] - (n/2-1)*a[n-4] + (n-1)*a[n-2]]];
%t Table[a[n], {n, 2, 40}] (* _Jean-François Alcover_, Aug 28 2022, after _Robert Israel_ *)
%K nonn
%O 2,3
%A _R. H. Hardin_, Feb 09 2009
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