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

Author

Gus Wiseman, Dec 03 2022

Keywords

• nonn

Comments

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

Examples

			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)
		

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.

Programs

Mathematica

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

Formula

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