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Regular triangle where T(n,k) is the number of finite multisets of positive integers with product n and sum k.
18

%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