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.

1, 4, 11, 35, 113, 326, 985, 3124, 8523, 24519, 71096, 191940, 530167, 1442059, 3833007, 10243259, 27151086, 71032191, 184492464, 478339983, 1227208513, 3140958369, 8016016201, 20210235189, 50962894061, 127936646350, 319022819270, 794501931062, 1969154638217

Offset

0,2

Links

Table of n, a(n) for n=0..28.

Gus Wiseman, Sequences enumerating triangles of integer partitions

Example

The a(1) = 1 through a(5) = 11 twice-partitions:
  (1)  (3)        (5)
       (21)       (32)
       (111)      (41)
       (1)(1)(1)  (221)
                  (311)
                  (2111)
                  (11111)
                  (3)(1)(1)
                  (21)(1)(1)
                  (111)(1)(1)
                  (1)(1)(1)(1)(1)

Mathematica

twiptn[n_]:=Join@@Table[Tuples[IntegerPartitions/@ptn], {ptn, IntegerPartitions[n]}];
Table[Length[Select[twiptn[n], OddQ[Length[#]]&&OddQ[Times@@Total/@#]&]], {n, 1, 15, 2}]

Crossrefs

For odd parts instead of length and sums we have A270995.

Requiring odd lengths and odd parts gives A279374 aerated.

This is the case of A358824 with odd sums.

This is the odd-length case (hence odd bisection) of A358825.

For odd lengths (instead of length) we have A358827.

For odd lengths instead of sums we have A358834.

A000009 counts partitions into odd parts.

A027193 counts partitions of odd length.

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

A078408 counts odd-length partitions into odd parts.

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

Cf. A000041, A001970, A072233, A271619, A279785, A356932.

Sequence in context: A052550 A197626 A349185 * A212910 A339034 A114248

Adjacent sequences: A358823 A358824 A358825 * A358827 A358828 A358829

Keyword

nonn

Author

Gus Wiseman, Dec 03 2022

Status

approved