A318537 - OEIS

%I #15 Jul 02 2022 16:09:23

%S 1,0,1,0,0,6,6,0,0,6,174,840,2520,5040,5040,0,0,0,840,24360,335160,

%T 3553200,32382000,259459200,1816214400,10897286400,54486432000,

%U 217945728000,653837184000,1307674368000,1307674368000,0,0,0,2520,335160,15198120,476496720,12767000400,314181504000,7288444800000

%N Irregular triangle read by rows: T(n,m) is the number of n X m (0,1)-matrices with pairwise distinct nonzero columns and pairwise distinct nonzero rows, n >= 0, m = 0..2^n-1.

%C T(n,m) is divisible by both n! and m!, but not necessarily by n!*m!.

%C By symmetry T(n,m) = T(m,n).

%C T(n,2^n-1) = T(n,2^n-2) = (2^n-1)! = A028366(n).

%F T(n,m) = m! * Sum_{i=0..n} Stirling1(n+1,i+1) * binomial(2^i-1,m) = n! * Sum_{j=0..m} Stirling1(m+1,j+1) * binomial(2^j-1,n).

%F T(n,m) = A059202(n,m) * m!.

%e Triangle begins:

%e n=0: 1;

%e n=1: 0, 1;

%e n=2: 0, 0, 6, 6;

%e n=3: 0, 0, 6, 174, 840, 2520, 5040, 5040;

%e ...

%o (PARI) { A318537(n,m) = m! * sum(i=0,n, stirling(n+1,i+1)*binomial(2^i - 1,m)); }

%Y Cf. A318538 (main diagonal), A059202.

%K nonn,tabf

%O 0,6

%A _Max Alekseyev_, Aug 28 2018