%I #12 Nov 23 2020 13:02:27
%S 1,1,1,1,1,1,1,2,1,1,1,0,0,1,1,1,1,3,2,1,1,1,1,2,1,2,1,1,1,2,1,3,3,2,
%T 1,1,1,0,3,0,3,3,2,1,1,1,1,0,3,3,3,3,2,1,1,1,1,2,3,2,1,5,3,2,1,1,1,2,
%U 2,1,3,7,3,5,3,2,1,1,1,0,0,2,2,2,5,3,5,3,2,1,1,1,1,0,3,3,4,5,5,3,5,3
%N Triangle read by rows: T(n, k) is the number of integer multisets of size k (partitions of k) for which the number of partitions of n with matching multiplicity multiset is odd (n >= 1, 1 <= k <= n).
%C The relevant partitions of n have exactly k parts.
%C The number of multiplicity multisets of size k met by a positive even number of partitions of n is A337584(n, k) - T(n, k).
%H Álvar Ibeas, <a href="/A337586/b337586.txt">First 72 rows, flattened</a>
%H Álvar Ibeas, <a href="/A337586/a337586.txt">First 30 rows</a>
%F T(n, k) == A008284(n, k) (mod 2).
%F If k > (2*n+1)/3, T(n, k) = A337585(n - k).
%e The 3 = A008284(6, 2) partitions of 6 into 2 parts show 2 = A337584(6, 2) different multiplicity multisets: (1, 1) is attained by two of those partitions ((5, 1) and (4, 2)) and the other (2) just by one, (3, 3). Then, T(6, 2) = 1.
%e Triangle begins:
%e k: 1 2 3 4 5 6 7 8 9 10
%e --------------------
%e n=1: 1
%e n=2: 1 1
%e n=3: 1 1 1
%e n=4: 1 2 1 1
%e n=5: 1 0 0 1 1
%e n=6: 1 1 3 2 1 1
%e n=7: 1 1 2 1 2 1 1
%e n=8: 1 2 1 3 3 2 1 1
%e n=9: 1 0 3 0 3 3 2 1 1
%e n=10: 1 1 0 3 3 3 3 2 1 1
%Y Cf. A008284, A337585 (row sums), A337584.
%K nonn,tabl
%O 1,8
%A _Álvar Ibeas_, Sep 02 2020