OFFSET
0,2
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)
LINKS
mjqxxxx, Proof of conjectured formulas, Mathematics Stack Exchange.
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
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Torlach Rush, Jul 27 2020
EXTENSIONS
More terms from Peter Luschny, Jun 05 2021
STATUS
approved