A283654 Triangle read by rows: number of n X m binary matrices with no rows or columns in which all entries are the same (n >= 1, 1 <= m <= n).

0, 0, 2, 0, 6, 102, 0, 14, 906, 22874, 0, 30, 6510, 417810, 17633670, 0, 62, 42666, 6644714, 622433730, 46959933962, 0, 126, 267582, 99044946, 20218802310, 3204360965106, 451575174961302, 0, 254, 1641786, 1430529674, 630917888610, 208308918928634, 60134626974122946, 16271255119687320314

Offset

1,3

Links

Table of n, a(n) for n=1..36.

Formula

T(n,m) = T(m,n) = 2*A183109(n,m) + 2^(n*m) + (2^n-2)^m + (2^m-2)^n - 2*(2^m-1)^n - 2*(2^n-1)^m.

T(n,1)=0, T(n,2)=2^n-2, T(n,3)=6^n-6*(3^n-2^n).

Example

The T(2,3)=6 matrices are
1 0 1
0 1 0
and the matrices obtained by permutations of rows and columns.
First values in triangle
0;
0, 2;
0, 6, 102;
0, 14, 906, 22874;
0, 30, 6510, 417810, 17633670;
0, 62, 42666, 6644714, 622433730, 46959933962;
0, 126, 267582, 99044946, 20218802310, 3204360965106, 451575174961302;

Maple

T0:=(n, m)->add((-1)^(m+k)*binomial(n, k)*(2^k-1)^m, k=0..n):
T:=(n, m)->2*T0(n, m)+2^(n*m)+(2^n-2)^m+(2^m-2)^n-2*(2^m-1)^n-2*(2^n-1)^m:
seq(seq(T(n, m), m=1..n), n=1..10);

Mathematica

T[n_, m_] := Sum[(-1)^j*Binomial[m, j]*(2^(m - j) - 1)^n, {j, 0, m}]; Flatten[Table[2*T[n, m] + 2^(n*m) + (2^n - 2)^m + (2^m - 2)^n - 2*(2^m - 1)^n - 2*(2^n - 1)^m, {n, 10}, {m, n}]] (* Indranil Ghosh, Mar 14 2017 *)

Prog

(PARI) T(n, m) = sum(j=0, m, (-1)^j*binomial(m, j)*(2^(m - j) - 1)^n);
tabl(nn) = {for(n=1, nn, for(m=1, n, print1(2*T(n, m) + 2^(n*m) + (2^n - 2)^m + (2^m - 2)^n - 2*(2^m - 1)^n - 2*(2^n - 1)^m, ", "); ); print(); ); };
tabl(10); \\ Indranil Ghosh, Mar 14 2017
(Python)
import math
f=math.factorial
def C(n, r): return f(n)/f(r)/f(n - r)
def T(n, m): return sum([(-1)**j*C(m, j)*(2**(m - j) - 1)**n for j in range (0, m+1)])
i=1
for n in range(1, 11):
....for m in range(1, n+1):
........print str(i)+" "+str(2*T(n, m) + 2**(n*m) + (2**n - 2)**m + (2**m - 2)**n - 2*(2**m - 1)**n - 2*(2**n - 1)**m)
........i+=1 # Indranil Ghosh, Mar 14 2017

Crossrefs

Diagonal gives A283624.

Cf. A183109.

Sequence in context: A196354 A305620 A294470 * A291373 A342700 A021487

Adjacent sequences: A283651 A283652 A283653 * A283655 A283656 A283657

Keyword

nonn,tabl

Author

Robert FERREOL, Mar 14 2017

Status

approved