cp's OEIS Frontend

A358826 Number of ways to choose a sequence of partitions, one of each part of an odd-length partition of 2n+1 into odd parts.

%I A358826 #6 Dec 04 2022 08:33:36
%S A358826 1,4,11,35,113,326,985,3124,8523,24519,71096,191940,530167,1442059,
%T A358826 3833007,10243259,27151086,71032191,184492464,478339983,1227208513,
%U A358826 3140958369,8016016201,20210235189,50962894061,127936646350,319022819270,794501931062,1969154638217
%N A358826 Number of ways to choose a sequence of partitions, one of each part of an odd-length partition of 2n+1 into odd parts.
%H A358826 Gus Wiseman, <a href="/A063834/a063834.txt">Sequences enumerating triangles of integer partitions</a>
%e A358826 The a(1) = 1 through a(5) = 11 twice-partitions:
%e A358826   (1)  (3)        (5)
%e A358826        (21)       (32)
%e A358826        (111)      (41)
%e A358826        (1)(1)(1)  (221)
%e A358826                   (311)
%e A358826                   (2111)
%e A358826                   (11111)
%e A358826                   (3)(1)(1)
%e A358826                   (21)(1)(1)
%e A358826                   (111)(1)(1)
%e A358826                   (1)(1)(1)(1)(1)
%t A358826 twiptn[n_]:=Join@@Table[Tuples[IntegerPartitions/@ptn],{ptn,IntegerPartitions[n]}];
%t A358826 Table[Length[Select[twiptn[n],OddQ[Length[#]]&&OddQ[Times@@Total/@#]&]],{n,1,15,2}]
%Y A358826 For odd parts instead of length and sums we have A270995.
%Y A358826 Requiring odd lengths and odd parts gives A279374 aerated.
%Y A358826 This is the case of A358824 with odd sums.
%Y A358826 This is the odd-length case (hence odd bisection) of A358825.
%Y A358826 For odd lengths (instead of length) we have A358827.
%Y A358826 For odd lengths instead of sums we have A358834.
%Y A358826 A000009 counts partitions into odd parts.
%Y A358826 A027193 counts partitions of odd length.
%Y A358826 A063834 counts twice-partitions, strict A296122, row-sums of A321449.
%Y A358826 A078408 counts odd-length partitions into odd parts.
%Y A358826 A300301 aerated counts twice-partitions with odd sums and parts.
%Y A358826 Cf. A000041, A001970, A072233, A271619, A279785, A356932.
%K A358826 nonn
%O A358826 0,2
%A A358826 _Gus Wiseman_, Dec 03 2022