This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A336174 #77 Jul 27 2022 09:04:48
%S A336174 0,0,0,2,16,80,360,1680,8064,39872,209920,1168640,6779520,41403648,
%T A336174 265434624,1765487360,12227461120,88163164160,656547803136,
%U A336174 5054718763008,40261284495360,330010833797120,2783003768258560,24166721457815552,215318925878132736,1966855934150246400
%N A336174 Number of non-symmetric binary n X n matrices M over the reals such that M^2 is the transpose of M.
%C A336174 We classify the (0,1) n X n matrices M_n by k, the number of 1's.
%C A336174 Let [T(n,k), n >= 0, k=0..n], be the lower triangular matrix where T(n,k) is the number of M^2 matrices equal to the transpose of M for n and k. Then:
%C A336174 T(n,n) = A001471(n).
%C A336174 Column sequences k=3..7 (without leading 0's) are:
%C A336174 T(n,3) = A001471(3) * A000292(n+1).
%C A336174 T(n,4) = A001471(4) * A000332(n+4).
%C A336174 T(n,5) = A001471(5) * A000389(n+5).
%C A336174 T(n,6) = A001471(6) * A000579(n+6).
%C A336174 T(n,7) = A001471(7) * A000580(n+7).
%C A336174 Row sums of T(n,k) generate known terms of this sequence and the next term a(10) evaluates to 209920 (see conjectured formula below).
%F A336174 a(n) = Sum_{k=0..n} A001471(k) * binomial(n, k). [Previously conjectured, for a proof see the link in A344912.]
%F A336174 From _Peter Luschny_, Jun 05 2021: (Start)
%F A336174 a(n) = 2^n*(add(n!/(24^k * k! * (n - 3*k)!), k=0..n/3) - 1).
%F A336174 a(n) = 2^n*(hypergeom([-n/3, (1 - n)/3, (2 - n)/3], [], -9/8) - 1).
%F A336174 a(n) = [x^n] exp(x*(x^2 + 6)/3) - exp(2*x). (End)
%F A336174 D-finite with recurrence (-n+3)*a(n) +4*(n-2)*a(n-1) +4*(-n+1)*a(n-2) +(n-1)*(n-2)*(n-3)*a(n-3) -2*(n-1)*(n-2)*(n-3)*a(n-4)=0. - _R. J. Mathar_, Jul 27 2022
%e A336174 a(3) = 2 because [0,1,0] [0,1,0] [0,0,1]
%e A336174 [0,0,1] * [0,0,1] = [1,0,0]
%e A336174 [1,0,0] [1,0,0] [0,1,0],
%e A336174 and [0,0,1] [0,0,1] [0,1,0]
%e A336174 [1,0,0] * [1,0,0] = [0,0,1]
%e A336174 [0,1,0] [0,1,0] [1,0,0].
%p A336174 a := n -> 2^n*(add(n!/(24^k*k!*(n-3*k)!), k=0..n/3) - 1): seq(a(n), n=0..25);
%p A336174 # Alternative:
%p A336174 gf := exp(x*(x^2+6)/3) - exp(2*x): ser := series(gf,x,32):
%p A336174 seq(n!*coeff(ser,x,n), n = 0..25); # _Peter Luschny_, Jun 05 2021
%o A336174 (PARI) m(n, t) = matrix(n, n, i, j, (t>>(i*n+j-n-1))%2)
%o A336174 a(n) = sum(t = 0, 2^n^2-1, m(n, t)^2 == m(n, t)~) - 2^n
%o A336174 for(n = 0, 9, print1(a(n), ", "))
%Y A336174 Cf. A000292, A000332, A000389, A000579, A000580, A001471, A344912.
%K A336174 nonn
%O A336174 0,4
%A A336174 _Torlach Rush_, Jul 10 2020
%E A336174 More terms from _Peter Luschny_, Jun 05 2021