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A331485 Array read by antidiagonals: A(n,k) is the number of nonequivalent nonnegative integer matrices with k columns and any number of nonzero rows with column sums n up to permutation of rows and columns.

%I A331485 #10 Feb 01 2020 22:41:05
%S A331485 1,1,1,1,1,1,1,2,2,1,1,3,7,3,1,1,5,23,21,5,1,1,7,79,162,66,7,1,1,11,
%T A331485 274,1636,1338,192,11,1,1,15,1003,19977,43686,10585,565,15,1,1,22,
%U A331485 3763,298416,2142277,1178221,82694,1579,22,1,1,30,14723,5300296,149056260,232984145,30370346,612700,4348,30,1
%N A331485 Array read by antidiagonals: A(n,k) is the number of nonequivalent nonnegative integer matrices with k columns and any number of nonzero rows with column sums n up to permutation of rows and columns.
%C A331485 A(n,k) is the number of non-isomorphic multiset partitions (multisets of multisets) with k parts each of size n.
%H A331485 Andrew Howroyd, <a href="/A331485/b331485.txt">Table of n, a(n) for n = 0..152</a>
%F A331485 A306017(n) = Sum_{d|n} A(n/d, d).
%e A331485 Array begins:
%e A331485 ============================================================
%e A331485 n\k | 0  1   2     3        4           5              6
%e A331485 ----+-------------------------------------------------------
%e A331485   0 | 1  1   1     1        1           1              1 ...
%e A331485   1 | 1  1   2     3        5           7             11 ...
%e A331485   2 | 1  2   7    23       79         274           1003 ...
%e A331485   3 | 1  3  21   162     1636       19977         298416 ...
%e A331485   4 | 1  5  66  1338    43686     2142277      149056260 ...
%e A331485   5 | 1  7 192 10585  1178221   232984145    74676589469 ...
%e A331485   6 | 1 11 565 82694 30370346 23412296767 33463656939910 ...
%e A331485   ...
%e A331485 The A(2,2) = 7 matrices are:
%e A331485   [1 0]  [2 0]  [1 1]  [2 1]  [2 0]  [1 1]  [2 2]
%e A331485   [1 0]  [0 1]  [1 0]  [0 1]  [0 2]  [1 1]
%e A331485   [0 1]  [0 1]  [0 1]
%e A331485   [0 1]
%o A331485 (PARI) \\ See A318951 for RowSumMats
%o A331485 T(n, k)={RowSumMats(k, n*k, n)}
%o A331485 { for(n=0, 7, for(k=0, 6, print1(T(n, k), ", ")); print) }
%Y A331485 Rows n=0..4 are A000012, A000041, A007717, A058194, A331721.
%Y A331485 Columns k=0..3 are A000012, A000041, A331722, A331723.
%Y A331485 Cf. A219727, A306017, A316674, A318951, A331315, A331461.
%K A331485 nonn,tabl
%O A331485 0,8
%A A331485 _Andrew Howroyd_, Jan 18 2020