A283624 Number of {0,1} n X n matrices with no rows or columns in which all entries are the same.

1, 0, 2, 102, 22874, 17633670, 46959933962, 451575174961302, 16271255119687320314, 2253375946574190518740230, 1219041140314101911449662059402, 2601922592659455476330065914740044182, 22040870572750372076278589658097827953983034

Offset: 0

Author

Robert FERREOL, Mar 12 2017

Keywords

• nonn

Comments

Every row and column must contain both a 0 and a 1 .

a(n) is the number of relations on n labeled points such that for every point x there exists y,z,t,u such that xRy, zRx, not(xRt), and not(uRx).

Examples

			For n=2 the a(2)=2 matrices are
  0 1
  1 0
and
  1 0
  0 1
		

Crossrefs

Cf. A048291.

Diagonal of A283654.

Programs

Maple

seq(2*sum((-1)^(n+k)*binomial(n,k)*(2^k-1)^n,k=0..n)+2^(n^2)+2*(2^n-2)^n-4*(2^n-1)^n,n=0..10)

Mathematica

Table[If[n==0, 1, 2 Sum[(-1)^(n + k) * Binomial[n, k] * (2^k - 1)^n, {k, 0,n}] + 2^(n^2) + 2*(2^n - 2)^n - 4*(2^n - 1)^n], {n, 0, 12}] (* Indranil Ghosh, Mar 12 2017 *)

PARI

for(n=0, 12, print1(2*sum(k=0, n, (-1)^(n + k) * binomial(n, k) * (2^k - 1)^n) + 2^(n^2) + 2*(2^n - 2)^n - 4*(2^n - 1)^n,", ")) \\ Indranil Ghosh, Mar 12 2017

Python

import math
f = math.factorial
def C(n, r): return f(n)//f(r)//f(n - r)
def A(n):
    s=0
    for k in range(0, n+1):
        s+=(-1)**(n + k) * C(n, k) * (2**k -1)**n
    return 2*s + 2**(n**2) + 2*(2**n - 2)**n - 4*(2**n - 1)**n # Indranil Ghosh, Mar 12 2017

Formula

a(n) = 2*Sum_{k=0..n} ((-1)^(n+k)*binomial(n,k)*(2^k-1)^n) + 2^(n^2) + 2*(2^n-2)^n - 4*(2^n-1)^n.

a(n) = 2*A048291(n) + 2^(n^2) + 2*(2^n-2)^n - 4*(2^n-1)^n.

Extensions

a(11)-a(12) from Indranil Ghosh, Mar 12 2017