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%I #11 Dec 23 2020 01:50:50
%S 1,9,29,68,134,237,388,600,887,1265,1751,2364,3124,4053,5174,6512,
%T 8093,9945,12097,14580,17426,20669,24344,28488,33139,38337,44123,
%U 50540,57632,65445,74026,83424,93689,104873,117029,130212,144478,159885,176492,194360,213551
%N Number of binary matrices with 3 distinct columns and any number of nonzero rows with n ones in every column and rows in nonincreasing lexicographic order.
%C The condition that the rows be in nonincreasing order is equivalent to considering nonequivalent matrices up to permutation of rows.
%C a(n) is the number of T_0 n-regular set multipartitions (multisets of sets) on a 3-set.
%H Andrew Howroyd, <a href="/A331390/b331390.txt">Table of n, a(n) for n = 1..1000</a>
%F a(n) = round(((n+2)/2)^4) - 3*(n+1) + 2.
%e The a(2) = 9 matrices are:
%e [1, 0, 0] [1, 1, 0] [1, 0, 1] [1, 0, 0]
%e [1, 0, 0] [1, 0, 0] [1, 0, 0] [1, 0, 0]
%e [0, 1, 0] [0, 1, 0] [0, 1, 0] [0, 1, 1]
%e [0, 1, 0] [0, 0, 1] [0, 1, 0] [0, 1, 0]
%e [0, 0, 1] [0, 0, 1] [0, 0, 1] [0, 0, 1]
%e [0, 0, 1]
%e .
%e [1, 1, 1] [1, 1, 0] [1, 1, 0] [1, 0, 1] [1, 1, 0]
%e [1, 0, 0] [1, 0, 1] [1, 0, 0] [1, 0, 0] [1, 0, 1]
%e [0, 1, 0] [0, 1, 0] [0, 1, 1] [0, 1, 1] [0, 1, 1]
%e [0, 0, 1] [0, 0, 1] [0, 0, 1] [0, 1, 0]
%o (PARI) a(n) = {round(((n+2)/2)^4) - 3*(n+1) + 2}
%Y Column k=3 of A331126.
%K nonn
%O 1,2
%A _Andrew Howroyd_, Jan 15 2020
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