A358828 Number of twice-partitions of n with no singletons.

1, 0, 1, 2, 5, 8, 19, 30, 68, 111, 229, 380, 799, 1280, 2519, 4325, 8128, 13666, 25758, 43085, 79300, 134571, 240124, 407794, 730398, 1224821, 2152122, 3646566, 6338691, 10657427, 18469865, 30913539, 53108364, 88953395, 151396452, 253098400, 429416589

Offset

0,4

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..36.

Formula

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

Example

The a(2) = 1 through a(6) = 19 twice-partitions:
  (11)  (21)   (22)      (32)       (33)
        (111)  (31)      (41)       (42)
               (211)     (221)      (51)
               (1111)    (311)      (222)
               (11)(11)  (2111)     (321)
                         (11111)    (411)
                         (21)(11)   (2211)
                         (111)(11)  (3111)
                                    (21111)
                                    (111111)
                                    (21)(21)
                                    (22)(11)
                                    (31)(11)
                                    (111)(21)
                                    (21)(111)
                                    (211)(11)
                                    (111)(111)
                                    (1111)(11)
                                    (11)(11)(11)

Mathematica

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

Crossrefs

The version for multiset partitions of integer partitions is A304966.

Allowing singletons other than (1) gives A358829.

A002865 counts partitions with no 1's.

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

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

Sequence in context: A032063 A320426 A133147 * A082630 A025078 A218937

Adjacent sequences: A358825 A358826 A358827 * A358829 A358830 A358831

Keyword

nonn

Author

Gus Wiseman, Dec 03 2022

Status

approved