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A336614 Number of n X n (0,1)-matrices A over the reals such that A^2 is the transpose of A.

%I A336614 #35 Jun 07 2021 05:48:23
%S A336614 1,2,4,10,32,112,424,1808,8320,40384,210944,1170688,6783616,41411840,
%T A336614 265451008,1765520128,12227526656,88163295232,656548065280,
%U A336614 5054719287296,40261285543936,330010835894272,2783003772452864,24166721466204160,215318925894909952,1966855934183800832
%N A336614 Number of n X n (0,1)-matrices A over the reals such that A^2 is the transpose of A.
%C A336614 From _Peter Luschny_, Jun 04 2021: (Start)
%C A336614 a(n) = n! * [x^n] exp(x*(x^2 + 6)/3).
%C A336614 a(n) = 2*a(n - 1) + (n^2 - 3*n + 2)*a(n - 3) for n >= 3.
%C A336614 a(n) = Sum_{k=0..n/3} (2^(n-3*k)*n!)/(3^k*k!*(n-3*k)!).
%C A336614 a(n) = 2^n*hypergeom([-n/3, (1-n)/3, (2-n)/3], [], -9/8).
%C A336614 [The above formulas, first stated as conjectures, were proved by mjqxxxx at Mathematics Stack Exchange, see link.] (End)
%H A336614 <a href="https://math.stackexchange.com/users/5546/mjqxxxx">mjqxxxx</a>, <a href="https://math.stackexchange.com/q/4164050">Proof of conjectured formulas</a>, Mathematics Stack Exchange.
%F A336614 a(n) = A336174(n) + A000079(n).
%e A336614 a(3) = A336174(3) + A000079(3) = 2 + 8 = 10.
%p A336614 a := n -> add((2^(n - 3*k)*n!)/(3^k*k!*(n - 3*k)!), k=0..n/3):
%p A336614 seq(a(n), n=0..25); # _Peter Luschny_, Jun 05 2021
%o A336614 (PARI) m(n, t) = matrix(n, n, i, j, (t>>(i*n+j-n-1))%2)
%o A336614 a(n) = sum(t = 0, 2^n^2-1, m(n, t)^2 == m(n, t)~)
%o A336614 for(n = 0, 9, print1(a(n), ", "))
%o A336614 (Python)
%o A336614 from itertools import product
%o A336614 from sympy import Matrix
%o A336614 def A336614(n):
%o A336614     c = 0
%o A336614     for d in product((0,1),repeat=n*n):
%o A336614         M = Matrix(d).reshape(n,n)
%o A336614         if M*M == M.T:
%o A336614             c += 1
%o A336614     return c # _Chai Wah Wu_, Sep 29 2020
%Y A336614 Row sums of A344912.
%Y A336614 Cf. A000079, A001471, A336174.
%K A336614 nonn
%O A336614 0,2
%A A336614 _Torlach Rush_, Jul 27 2020
%E A336614 More terms from _Peter Luschny_, Jun 05 2021