A319000 Regular triangle where T(n,k) is the number of finite multisets of positive integers with product n and sum k.

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, 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, 0, 0, 0, 0, 2, 3, 3, 3, 3, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset

1,10

Links

Table of n, a(n) for n=1..87.

Example

Triangle begins:
  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 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 0 0 0 0 2 3 3 3 3 4
  0 0 0 0 0 0 0 0 0 0 0 0 1
  0 0 0 0 0 0 0 0 1 1 1 1 1 2
  0 0 0 0 0 0 0 1 1 1 1 1 1 1 2
  0 0 0 0 0 0 0 3 3 4 4 4 4 4 4 5
  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1
  0 0 0 0 0 0 0 1 2 2 3 3 3 3 3 3 3 4
  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1
  0 0 0 0 0 0 0 0 2 2 2 3 3 3 3 3 3 3 3 4
Row 12 {0,0,0,0,0,0,2,3,3,3,3,4} corresponds to the partitions (C = 12):
. . . . . . (43)  (62)   (621)   (6211)   (62111)    (C)
            (322) (431)  (4311)  (43111)  (431111)   (621111)
                  (3221) (32211) (322111) (3221111)  (4311111)
                                          (32211111)

Mathematica

Table[Length[Select[IntegerPartitions[k], Times@@#==n&]], {n, 20}, {k, n}]

Crossrefs

Row sums are A319916. Column sums are A319005. Last column is A001055.

Cf. A000041, A002865, A069016, A096276, A301987, A318950, A319057.

Sequence in context: A112765 A105966 A318950 * A083915 A083892 A354451

Adjacent sequences: A318997 A318998 A318999 * A319001 A319002 A319003

Keyword

nonn,tabl

Author

Gus Wiseman, Oct 22 2018

Status

approved