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A318537 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.

%I A318537 #15 Jul 02 2022 16:09:23
%S A318537 1,0,1,0,0,6,6,0,0,6,174,840,2520,5040,5040,0,0,0,840,24360,335160,
%T A318537 3553200,32382000,259459200,1816214400,10897286400,54486432000,
%U A318537 217945728000,653837184000,1307674368000,1307674368000,0,0,0,2520,335160,15198120,476496720,12767000400,314181504000,7288444800000
%N A318537 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 A318537 T(n,m) is divisible by both n! and m!, but not necessarily by n!*m!.
%C A318537 By symmetry T(n,m) = T(m,n).
%C A318537 T(n,2^n-1) = T(n,2^n-2) = (2^n-1)! = A028366(n).
%F A318537 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 A318537 T(n,m) = A059202(n,m) * m!.
%e A318537 Triangle begins:
%e A318537 n=0: 1;
%e A318537 n=1: 0, 1;
%e A318537 n=2: 0, 0, 6, 6;
%e A318537 n=3: 0, 0, 6, 174, 840, 2520, 5040, 5040;
%e A318537 ...
%o A318537 (PARI) { A318537(n,m) = m! * sum(i=0,n, stirling(n+1,i+1)*binomial(2^i - 1,m)); }
%Y A318537 Cf. A318538 (main diagonal), A059202.
%K A318537 nonn,tabf
%O A318537 0,6
%A A318537 _Max Alekseyev_, Aug 28 2018