A283654 - OEIS

%I #52 Mar 25 2017 10:23:24

%S 0,0,2,0,6,102,0,14,906,22874,0,30,6510,417810,17633670,0,62,42666,

%T 6644714,622433730,46959933962,0,126,267582,99044946,20218802310,

%U 3204360965106,451575174961302,0,254,1641786,1430529674,630917888610,208308918928634,60134626974122946,16271255119687320314

%N 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).

%F 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.

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

%e The T(2,3)=6 matrices are

%e 1 0 1

%e 0 1 0

%e and the matrices obtained by permutations of rows and columns.

%e First values in triangle

%e 0;

%e 0, 2;

%e 0, 6, 102;

%e 0, 14, 906, 22874;

%e 0, 30, 6510, 417810, 17633670;

%e 0, 62, 42666, 6644714, 622433730, 46959933962;

%e 0, 126, 267582, 99044946, 20218802310, 3204360965106, 451575174961302;

%p T0:=(n,m)->add((-1)^(m+k)*binomial(n,k)*(2^k-1)^m, k=0..n):

%p 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:

%p seq(seq(T(n,m), m=1..n),n=1..10);

%t 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 *)

%o (PARI) T(n, m) = sum(j=0, m, (-1)^j*binomial(m, j)*(2^(m - j) - 1)^n);

%o 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(););};

%o tabl(10); \\ _Indranil Ghosh_, Mar 14 2017

%o (Python)

%o import math

%o f=math.factorial

%o def C(n,r): return f(n)/f(r)/f(n - r)

%o def T(n,m): return sum([(-1)**j*C(m,j)*(2**(m - j) - 1)**n for j in range (0, m+1)])

%o i=1

%o for n in range(1,11):

%o ....for m in range(1, n+1):

%o ........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)

%o ........i+=1 # _Indranil Ghosh_, Mar 14 2017

%Y Diagonal gives A283624.

%Y Cf. A183109.

%K nonn,tabl

%O 1,3

%A _Robert FERREOL_, Mar 14 2017