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

0,3

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).

Links

Table of n, a(n) for n=0..12.

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.

Example

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

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 *)

Prog

(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

Crossrefs

Cf. A048291.

Diagonal of A283654.

Sequence in context: A256516 A157066 A202740 * A055693 A357767 A215964

Adjacent sequences: A283621 A283622 A283623 * A283625 A283626 A283627

Keyword

nonn

Author

Robert FERREOL, Mar 12 2017

Extensions

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

Status

approved