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%I #5 Dec 03 2022 13:54:02
%S 1,0,1,2,5,8,19,30,68,111,229,380,799,1280,2519,4325,8128,13666,25758,
%T 43085,79300,134571,240124,407794,730398,1224821,2152122,3646566,
%U 6338691,10657427,18469865,30913539,53108364,88953395,151396452,253098400,429416589
%N Number of twice-partitions of n with no singletons.
%C A twice-partition of n is a sequence of integer partitions, one of each part of an integer partition of n.
%F G.f.: Product_{k>=1} 1/(1-(A000041(k)-1)*x^k).
%e The a(2) = 1 through a(6) = 19 twice-partitions:
%e (11) (21) (22) (32) (33)
%e (111) (31) (41) (42)
%e (211) (221) (51)
%e (1111) (311) (222)
%e (11)(11) (2111) (321)
%e (11111) (411)
%e (21)(11) (2211)
%e (111)(11) (3111)
%e (21111)
%e (111111)
%e (21)(21)
%e (22)(11)
%e (31)(11)
%e (111)(21)
%e (21)(111)
%e (211)(11)
%e (111)(111)
%e (1111)(11)
%e (11)(11)(11)
%t twiptn[n_]:=Join@@Table[Tuples[IntegerPartitions/@ptn],{ptn,IntegerPartitions[n]}];
%t Table[Length[Select[twiptn[n],FreeQ[Length/@#,1]&]],{n,0,10}]
%Y The version for multiset partitions of integer partitions is A304966.
%Y Allowing singletons other than (1) gives A358829.
%Y A002865 counts partitions with no 1's.
%Y A063834 counts twice-partitions, strict A296122, row-sums of A321449.
%Y Cf. A000009, A000041, A000219, A001970, A072233, A358824.
%K nonn
%O 0,4
%A _Gus Wiseman_, Dec 03 2022
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