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

1, 2, 4, 10, 32, 112, 424, 1808, 8320, 40384, 210944, 1170688, 6783616, 41411840, 265451008, 1765520128, 12227526656, 88163295232, 656548065280, 5054719287296, 40261285543936, 330010835894272, 2783003772452864, 24166721466204160, 215318925894909952, 1966855934183800832

Offset: 0

Author

Torlach Rush, Jul 27 2020

Keywords

• nonn

Comments

From Peter Luschny, Jun 04 2021: (Start)

a(n) = n! * [x^n] exp(x*(x^2 + 6)/3).

a(n) = 2*a(n - 1) + (n^2 - 3*n + 2)*a(n - 3) for n >= 3.

a(n) = Sum_{k=0..n/3} (2^(n-3*k)*n!)/(3^k*k!*(n-3*k)!).

a(n) = 2^n*hypergeom([-n/3, (1-n)/3, (2-n)/3], [], -9/8).

[The above formulas, first stated as conjectures, were proved by mjqxxxx at Mathematics Stack Exchange, see link.] (End)

Examples

			a(3) = A336174(3) + A000079(3) = 2 + 8 = 10.
		

Links

• mjqxxxx, Proof of conjectured formulas, Mathematics Stack Exchange.

Crossrefs

Row sums of A344912.

Cf. A000079, A001471, A336174.

Programs

Maple

a := n -> add((2^(n - 3*k)*n!)/(3^k*k!*(n - 3*k)!), k=0..n/3):
seq(a(n), n=0..25); # Peter Luschny, Jun 05 2021

PARI

m(n, t) = matrix(n, n, i, j, (t>>(i*n+j-n-1))%2)
a(n) = sum(t = 0, 2^n^2-1, m(n, t)^2 == m(n, t)~)
for(n = 0, 9, print1(a(n), ", "))

Python

from itertools import product
from sympy import Matrix
def A336614(n):
    c = 0
    for d in product((0,1),repeat=n*n):
        M = Matrix(d).reshape(n,n)
        if M*M == M.T:
            c += 1
    return c # Chai Wah Wu, Sep 29 2020

Formula

a(n) = A336174(n) + A000079(n).

Extensions

More terms from Peter Luschny, Jun 05 2021