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%I A358827 #6 Dec 04 2022 08:33:40
%S A358827 1,1,1,3,3,7,11,19,27,51,83,128,208,324,542,856,1332,2047,3371,5083,
%T A358827 8009,12545,19478,29770,46038,70777,108627,167847,255408,388751,
%U A358827 593475,901108,1361840,2077973,3125004,4729056,7146843,10732799,16104511,24257261,36305878
%N A358827 Number of twice-partitions of n into partitions with all odd lengths and sums.
%C A358827 A twice-partition of n is a sequence of integer partitions, one of each part of an integer partition of n.
%H A358827 Gus Wiseman, <a href="/A063834/a063834.txt">Sequences enumerating triangles of integer partitions</a>
%F A358827 G.f.: Product_{k odd} 1/(1-A027193(k)*x^k).
%e A358827 The a(1) = 1 through a(6) = 11 twice-partitions:
%e A358827   (1)  (1)(1)  (3)        (3)(1)        (5)              (3)(3)
%e A358827                (111)      (111)(1)      (221)            (5)(1)
%e A358827                (1)(1)(1)  (1)(1)(1)(1)  (311)            (111)(3)
%e A358827                                         (11111)          (221)(1)
%e A358827                                         (3)(1)(1)        (3)(111)
%e A358827                                         (111)(1)(1)      (311)(1)
%e A358827                                         (1)(1)(1)(1)(1)  (111)(111)
%e A358827                                                          (11111)(1)
%e A358827                                                          (3)(1)(1)(1)
%e A358827                                                          (111)(1)(1)(1)
%e A358827                                                          (1)(1)(1)(1)(1)(1)
%t A358827 twiptn[n_]:=Join@@Table[Tuples[IntegerPartitions/@ptn],{ptn,IntegerPartitions[n]}];
%t A358827 Table[Length[Select[twiptn[n],OddQ[Times@@Length/@#]&&OddQ[Times@@Total/@#]&]],{n,0,10}]
%Y A358827 This is the case of A358334 with odd sums.
%Y A358827 This is the case of A358825 with odd lengths.
%Y A358827 The case of odd length is the odd bisection.
%Y A358827 For odd parts instead of lengths and sums we have A270995.
%Y A358827 Requiring odd parts also gives A279374 aerated.
%Y A358827 A000009 counts partitions into odd parts.
%Y A358827 A027193 counts partitions of odd length.
%Y A358827 A063834 counts twice-partitions, strict A296122, row-sums of A321449.
%Y A358827 A078408 counts odd-length partitions into odd parts.
%Y A358827 A300301 aerated counts twice-partitions with odd sums and parts.
%Y A358827 Cf. A000041, A001970, A072233, A271619, A279785, A306319, A356932, A358824.
%K A358827 nonn
%O A358827 0,4
%A A358827 _Gus Wiseman_, Dec 03 2022

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