A365067 - OEIS

A365067

Irregular triangle read by rows where T(n,k) is the number of integer partitions of n whose odd parts sum to k, for k ranging from mod(n,2) to n in steps of 2.

1, 1, 1, 1, 1, 2, 2, 1, 2, 2, 2, 3, 3, 2, 2, 4, 3, 4, 3, 5, 5, 3, 4, 4, 6, 5, 6, 6, 5, 8, 7, 5, 6, 8, 6, 10, 7, 10, 9, 10, 8, 12, 11, 7, 10, 12, 12, 10, 15, 11, 14, 15, 15, 16, 12, 18, 15, 11, 14, 20, 18, 20, 15, 22, 15, 22, 21, 25, 24, 24, 18, 27

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OFFSET

0,6

COMMENTS

The version for all k = 0..n is A113685 (including zeros).

LINKS

Table of n, a(n) for n=0..71.

FORMULA

T(n,k) = A000009(k) * A000041((n-k)/2).

EXAMPLE

Triangle begins:

1 1

1 2

2 1 2

2 2 3

3 2 2 4

3 4 3 5

5 3 4 4 6

5 6 6 5 8

7 5 6 8 6 10

7 10 9 10 8 12

11 7 10 12 12 10 15

11 14 15 15 16 12 18

15 11 14 20 18 20 15 22

15 22 21 25 24 24 18 27

Row n = 8 counts the following partitions:

(8) (611) (431) (521) (71)

(62) (4211) (41111) (332) (53)

(44) (22211) (3221) (32111) (5111)

(422) (221111) (2111111) (3311)

(2222) (311111)

(11111111)

Row n = 9 counts the following partitions:

(81) (63) (54) (72) (9)

(621) (6111) (522) (5211) (711)

(441) (432) (4311) (3321) (531)

(4221) (42111) (411111) (321111) (51111)

(22221) (3222) (32211) (21111111) (333)

(222111) (2211111) (33111)

(3111111)

(111111111)

MATHEMATICA

Table[Length[Select[IntegerPartitions[n], Total[Select[#, OddQ]]==k&]], {n, 0, 15}, {k, Mod[n, 2], n, 2}]

CROSSREFS

Row sums are A000041.

The version including all k is A113685, even version A113686.

Column k = 1 is A119620.

The even version and the reverse version are both A174713.

For odd-indexed instead of odd parts we have A346697, even version A346698.

The corresponding rank statistic is A366528, even version A366531.

A000009 counts partitions into odd parts, ranks A066208.

A086543 counts partitions with odd parts, ranks A366322.

A239261 counts partitions with (sum of odd parts) = (sum of even parts).

Cf. A035363, A045931, A053253, A066967, A130780, A171966, A241638, A268335, A325698, A366533.

Sequence in context: A168208 A333003 A352072 * A332086 A197081 A029395

Adjacent sequences: A365064 A365065 A365066 * A365068 A365069 A365070

KEYWORD

nonn,tabf

AUTHOR

Gus Wiseman, Oct 16 2023

STATUS

approved