%I #19 Oct 22 2018 17:42:07
%S 1,0,1,0,0,1,0,0,0,2,0,0,0,0,1,0,0,0,0,1,2,0,0,0,0,0,0,1,0,0,0,0,0,2,
%T 2,3,0,0,0,0,0,1,1,1,2,0,0,0,0,0,0,1,1,1,2,0,0,0,0,0,0,0,0,0,0,1,0,0,
%U 0,0,0,0,2,3,3,3,3,4,0,0,0,0,0,0,0,0,0
%N Regular triangle where T(n,k) is the number of finite multisets of positive integers with product n and sum k.
%e Triangle begins:
%e 1
%e 0 1
%e 0 0 1
%e 0 0 0 2
%e 0 0 0 0 1
%e 0 0 0 0 1 2
%e 0 0 0 0 0 0 1
%e 0 0 0 0 0 2 2 3
%e 0 0 0 0 0 1 1 1 2
%e 0 0 0 0 0 0 1 1 1 2
%e 0 0 0 0 0 0 0 0 0 0 1
%e 0 0 0 0 0 0 2 3 3 3 3 4
%e 0 0 0 0 0 0 0 0 0 0 0 0 1
%e 0 0 0 0 0 0 0 0 1 1 1 1 1 2
%e 0 0 0 0 0 0 0 1 1 1 1 1 1 1 2
%e 0 0 0 0 0 0 0 3 3 4 4 4 4 4 4 5
%e 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1
%e 0 0 0 0 0 0 0 1 2 2 3 3 3 3 3 3 3 4
%e 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1
%e 0 0 0 0 0 0 0 0 2 2 2 3 3 3 3 3 3 3 3 4
%e Row 12 {0,0,0,0,0,0,2,3,3,3,3,4} corresponds to the partitions (C = 12):
%e . . . . . . (43) (62) (621) (6211) (62111) (C)
%e (322) (431) (4311) (43111) (431111) (621111)
%e (3221) (32211) (322111) (3221111) (4311111)
%e (32211111)
%t Table[Length[Select[IntegerPartitions[k],Times@@#==n&]],{n,20},{k,n}]
%Y Row sums are A319916. Column sums are A319005. Last column is A001055.
%Y Cf. A000041, A002865, A069016, A096276, A301987, A318950, A319057.
%K nonn,tabl
%O 1,10
%A _Gus Wiseman_, Oct 22 2018