A358825 - OEIS

%I #7 Dec 04 2022 08:33:31

%S 1,1,1,4,4,11,20,35,56,113,207,326,602,985,1777,3124,5115,8523,15011,

%T 24519,41571,71096,115650,191940,320651,530167,865781,1442059,2358158,

%U 3833007,6325067,10243259,16603455,27151086,43734197,71032191,115091799,184492464

%N Number of ways to choose a sequence of integer partitions, one of each part of an integer partition of n into odd parts.

%F G.f.: Product_{k odd} 1/(1-A000041(k)*x^k).

%e The a(1) = 1 through a(5) = 11 twice-partitions:

%e   (1)  (1)(1)  (3)        (3)(1)        (5)

%e                (21)       (21)(1)       (32)

%e                (111)      (111)(1)      (41)

%e                (1)(1)(1)  (1)(1)(1)(1)  (221)

%e                                         (311)

%e                                         (2111)

%e                                         (11111)

%e                                         (3)(1)(1)

%e                                         (21)(1)(1)

%e                                         (111)(1)(1)

%e                                         (1)(1)(1)(1)(1)

%t twiptn[n_]:=Join@@Table[Tuples[IntegerPartitions/@ptn],{ptn,IntegerPartitions[n]}];

%t Table[Length[Select[twiptn[n],OddQ[Times@@Total/@#]&]],{n,0,10}]

%Y For odd parts instead of sums we have A270995.

%Y For distinct instead of odd sums we have A271619.

%Y Requiring odd length, odd lengths, and odd parts gives A279374 aerated.

%Y For odd lengths instead of sums we have A358334.

%Y The odd-length case is A358826.

%Y A000009 counts partitions into odd parts.

%Y A027193 counts partitions of odd length.

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

%Y A078408 counts odd-length partitions into odd parts.

%Y A300301 aerated counts twice-partitions with odd sums and parts.

%Y Cf. A000041, A001970, A072233, A279785, A356932, A358824.

%K nonn

%O 0,4

%A _Gus Wiseman_, Dec 03 2022