A358829 Number of twice-partitions of n with no (1)'s.

1, 0, 2, 3, 9, 13, 38, 56, 144, 237, 524, 886, 1961, 3225, 6700, 11702, 23007, 39787, 77647, 133707, 254896, 442736, 820703, 1427446, 2630008, 4535330, 8224819, 14250148, 25513615, 43981753, 78252954, 134323368, 236900355, 406174046, 709886932, 1213934012

Offset

0,3

Comments

A twice-partition of n is a sequence of integer partitions, one of each part of an integer partition of n.

Links

Table of n, a(n) for n=0..35.

Formula

G.f.: Product_{k>=2} 1/(1-A000041(k)*x^k).

Example

The a(2) = 2 through a(5) = 13 twice-partitions:
  (2)   (3)    (4)       (5)
  (11)  (21)   (22)      (32)
        (111)  (31)      (41)
               (211)     (221)
               (1111)    (311)
               (2)(2)    (2111)
               (11)(2)   (3)(2)
               (2)(11)   (11111)
               (11)(11)  (21)(2)
                         (3)(11)
                         (111)(2)
                         (21)(11)
                         (111)(11)

Mathematica

twiptn[n_]:=Join@@Table[Tuples[IntegerPartitions/@ptn], {ptn, IntegerPartitions[n]}];
Table[Length[Select[twiptn[n], FreeQ[Total/@#, 1]&]], {n, 0, 10}]

Crossrefs

The version for multiset partitions of integer partitions is A317911.

Forbidding all singletons gives A358828.

A002865 counts partitions with no 1's.

A063834 counts twice-partitions, strict A296122, row-sums of A321449.

Cf. A000009, A000041, A000219, A001970, A072233, A304966, A358824.

Sequence in context: A101234 A233190 A269154 * A225792 A057293 A109662

Adjacent sequences: A358826 A358827 A358828 * A358830 A358831 A358832

Keyword

nonn

Author

Gus Wiseman, Dec 03 2022

Status

approved