A358194 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n with partial sums summing to k, where k ranges from n to n(n+1)/2.

1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 2, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1

Offset

0,77

Comments

The partial sums of a sequence (a, b, c, ...) are (a, a+b, a+b+c, ...).

Links

Table of n, a(n) for n=0..92.

Example

Triangle begins:
  1
  1
  1 1
  1 0 1 1
  1 0 1 1 0 1 1
  1 0 0 1 1 0 1 1 0 1 1
  1 0 0 1 1 1 1 0 1 1 0 1 1 0 1 1
  1 0 0 0 1 1 1 0 1 1 1 1 1 0 1 1 0 1 1 0 1 1
  1 0 0 0 1 1 1 1 0 1 1 1 2 1 0 1 1 1 1 1 0 1 1 0 1 1 0 1 1
For example, the T(15,59) = 5 partitions are: (8,2,2,2,1), (7,3,3,1,1), (6,5,2,1,1), (4,3,2,2,2,2), (3,3,3,3,2,1).

Mathematica

Table[Length[Select[IntegerPartitions[n], Total[Accumulate[#]]==k&]], {n, 0, 8}, {k, n, n*(n+1)/2}]

Crossrefs

Row sums are A000041.

The version for compositions is A053632.

Row lengths are A152947.

The version for reversed partitions is A264034.

A048793 = partial sums of reversed standard compositions, sum A029931.

A358134 = partial sums of standard compositions, sum A359042.

A358136 = partial sums of prime indices, sum A318283.

A359361 = partial sums of reversed prime indices, sum A304818.

Cf. A000009, A325362, A358137, A359397.

Sequence in context: A212210 A127499 A198068 * A121361 A191907 A052343

Adjacent sequences: A358191 A358192 A358193 * A358195 A358196 A358197

Keyword

nonn,tabf

Author

Gus Wiseman, Dec 31 2022

Status

approved