A356932 -id:A356932 - cp's OEIS frontend

A358825 Number of ways to choose a sequence of integer partitions, one of each part of an integer partition of n into odd parts.

1, 1, 1, 4, 4, 11, 20, 35, 56, 113, 207, 326, 602, 985, 1777, 3124, 5115, 8523, 15011, 24519, 41571, 71096, 115650, 191940, 320651, 530167, 865781, 1442059, 2358158, 3833007, 6325067, 10243259, 16603455, 27151086, 43734197, 71032191, 115091799, 184492464

Offset: 0

Gus Wiseman, Dec 03 2022

nonn

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

For odd parts instead of sums we have A270995.
For distinct instead of odd sums we have A271619.
Requiring odd length, odd lengths, and odd parts gives A279374 aerated.
For odd lengths instead of sums we have A358334.
The odd-length case is A358826.
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, A279785, A356932, A358824.

twiptn[n_]:=Join@@Table[Tuples[IntegerPartitions/@ptn],{ptn,IntegerPartitions[n]}];
Table[Length[Select[twiptn[n],OddQ[Times@@Total/@#]&]],{n,0,10}]

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

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.

Original entry on oeis.org

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

Gus Wiseman, Dec 03 2022

nonn

			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)

Gus Wiseman, Sequences enumerating triangles of integer partitions

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.

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

A358827 Number of twice-partitions of n into partitions with all odd lengths and sums.

Original entry on oeis.org

1, 1, 1, 3, 3, 7, 11, 19, 27, 51, 83, 128, 208, 324, 542, 856, 1332, 2047, 3371, 5083, 8009, 12545, 19478, 29770, 46038, 70777, 108627, 167847, 255408, 388751, 593475, 901108, 1361840, 2077973, 3125004, 4729056, 7146843, 10732799, 16104511, 24257261, 36305878

Offset: 0

Gus Wiseman, Dec 03 2022

nonn

A twice-partition of n is a sequence of integer partitions, one of each part of an integer partition of n.

			The a(1) = 1 through a(6) = 11 twice-partitions:
  (1)  (1)(1)  (3)        (3)(1)        (5)              (3)(3)
               (111)      (111)(1)      (221)            (5)(1)
               (1)(1)(1)  (1)(1)(1)(1)  (311)            (111)(3)
                                        (11111)          (221)(1)
                                        (3)(1)(1)        (3)(111)
                                        (111)(1)(1)      (311)(1)
                                        (1)(1)(1)(1)(1)  (111)(111)
                                                         (11111)(1)
                                                         (3)(1)(1)(1)
                                                         (111)(1)(1)(1)
                                                         (1)(1)(1)(1)(1)(1)

Gus Wiseman, Sequences enumerating triangles of integer partitions

This is the case of A358334 with odd sums.
This is the case of A358825 with odd lengths.
The case of odd length is the odd bisection.
For odd parts instead of lengths and sums we have A270995.
Requiring odd parts also gives A279374 aerated.
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, A306319, A356932, A358824.

twiptn[n_]:=Join@@Table[Tuples[IntegerPartitions/@ptn],{ptn,IntegerPartitions[n]}];
Table[Length[Select[twiptn[n],OddQ[Times@@Length/@#]&&OddQ[Times@@Total/@#]&]],{n,0,10}]

G.f.: Product_{k odd} 1/(1-A027193(k)*x^k).

cp's OEIS Frontend

A358825 Number of ways to choose a sequence of integer partitions, one of each part of an integer partition of n into odd parts.

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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.

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Mathematica

A358827 Number of twice-partitions of n into partitions with all odd lengths and sums.

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Examples

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Programs

Mathematica

Formula