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
Keywords
Examples
For n=2 the a(2)=2 matrices are 0 1 1 0 and 1 0 0 1
Programs
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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)
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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
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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
Comments