%I #6 Jan 14 2020 15:28:10
%S 1,2,3,7,4,47,6,317,326,3730,13,78625,19,1372944,824798,36641157,39,
%T 1211620030,55,41615035330,9881046310,1743624029061,105,
%U 85034153219895,10679934643,4476101995508420,385900622506127,268621480352669227,257,17969848317035340096
%N Number of nonnegative integer matrices with total sum n, distinct columns with equal sums and any number of distinct nonzero rows in decreasing lexicographic order.
%C The condition that the rows be in decreasing order is equivalent to considering nonequivalent matrices with distinct rows up to permutation of rows.
%F a(n) = Sum_{d|n} A331160(n/d, d).
%F a(p) = A000009(n) + 1 for prime p.
%e The a(4) = 7 matrices are:
%e [1 0 0 0] [1 1] [2 1] [2 0] [1 2] [3] [4]
%e [0 1 0 0] [1 0] [0 1] [0 2] [1 0] [1]
%e [0 0 1 0] [0 1]
%e [0 0 0 1]
%Y Cf. A000009, A330158, A331160.
%K nonn
%O 1,2
%A _Andrew Howroyd_, Jan 13 2020