A379720 Except a(0)=1 and a(4)=0, number of integer partitions of n with no 1's and at least two parts.

1, 0, 0, 0, 0, 1, 3, 3, 6, 7, 11, 13, 20, 23, 33, 40, 54, 65, 87, 104, 136, 164, 209, 252, 319, 382, 477, 573, 707, 846, 1038, 1237, 1506, 1793, 2166, 2572, 3093, 3659, 4377, 5169, 6152, 7244, 8590, 10086, 11913, 13958, 16423, 19195, 22518, 26251, 30700, 35716

Offset

0,7

Comments

Also partitions of n such that all parts are > 1 and have product > n.

Allowing 1's gives A114324, ranks A325037. The strict case is A318029 (except first term).

Links

Table of n, a(n) for n=0..51.

Formula

Except for n = 0 and n = 4, we have a(n) = A002865(n) - 1.

Example

The a(5) = 1 through a(11) = 13 partitions:
  (3,2)  (3,3)    (4,3)    (4,4)      (5,4)      (5,5)        (6,5)
         (4,2)    (5,2)    (5,3)      (6,3)      (6,4)        (7,4)
         (2,2,2)  (3,2,2)  (6,2)      (7,2)      (7,3)        (8,3)
                           (3,3,2)    (3,3,3)    (8,2)        (9,2)
                           (4,2,2)    (4,3,2)    (4,3,3)      (4,4,3)
                           (2,2,2,2)  (5,2,2)    (4,4,2)      (5,3,3)
                                      (3,2,2,2)  (5,3,2)      (5,4,2)
                                                 (6,2,2)      (6,3,2)
                                                 (3,3,2,2)    (7,2,2)
                                                 (4,2,2,2)    (3,3,3,2)
                                                 (2,2,2,2,2)  (4,3,2,2)
                                                              (5,2,2,2)
                                                              (3,2,2,2,2)

Mathematica

Table[Length[Select[IntegerPartitions[n], FreeQ[#, 1]&&Plus@@#<Times@@#&]], {n, 0, 30}]

Crossrefs

For <= instead of < we have A002865 = partitions into parts > 1.

These partitions have ranks A071904 (except initial terms).

Set a(4) = 1 to get A083751.

A000041 counts integer partitions, strict A000009.

A379668 counts partitions without 1's by sum and product.

Counting and ranking multisets by comparing sum and product:

- same: A001055, ranks A301987

- divisible: A057567, ranks A326155

- divisor: A057568, ranks A326149, see A379733

- greater than: A096276 shifted right, ranks A325038

- greater or equal: A096276, ranks A325044

- less than: A114324, ranks A325037, see A318029

- less or equal: A319005, ranks A379721, see A025147

- different: A379736, ranks A379722, see A111133

Cf. A003963, A028422, A318950, A319000, A319916, A325036, A325041, A326152, A326178, A379666, A379678.

Sequence in context: A026926 A332557 A083751 * A034401 A345729 A350171

Adjacent sequences: A379717 A379718 A379719 * A379721 A379722 A379723

Keyword

nonn

Author

Gus Wiseman, Jan 06 2025

Status

approved