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A358829
Number of twice-partitions of n with no (1)'s.
1
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.
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.
Sequence in context: A101234 A233190 A269154 * A225792 A057293 A109662
KEYWORD
nonn
AUTHOR
Gus Wiseman, Dec 03 2022
STATUS
approved