cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A358824 Number of twice-partitions of n of odd length.

Original entry on oeis.org

0, 1, 2, 4, 7, 15, 32, 61, 121, 260, 498, 967, 1890, 3603, 6839, 12972, 23883, 44636, 82705, 150904, 275635, 501737, 905498, 1628293, 2922580, 5224991, 9296414, 16482995, 29125140, 51287098, 90171414, 157704275, 275419984, 479683837, 833154673, 1442550486, 2493570655
Offset: 0

Views

Author

Gus Wiseman, Dec 03 2022

Keywords

Comments

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

Examples

			The a(1) = 1 through a(5) = 15 twice-partitions:
  (1)  (2)   (3)        (4)         (5)
       (11)  (21)       (22)        (32)
             (111)      (31)        (41)
             (1)(1)(1)  (211)       (221)
                        (1111)      (311)
                        (2)(1)(1)   (2111)
                        (11)(1)(1)  (11111)
                                    (2)(2)(1)
                                    (3)(1)(1)
                                    (11)(2)(1)
                                    (2)(11)(1)
                                    (21)(1)(1)
                                    (11)(11)(1)
                                    (111)(1)(1)
                                    (1)(1)(1)(1)(1)

Links

Crossrefs

The version for set partitions is A024429.
For odd lengths (instead of length) we have A358334.
The case of odd parts also is A358823.
The case of odd sums also is A358826.
The case of odd lengths also is A358834.
For multiset partitions of integer partitions: A358837, ranked by A026424.
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, A270995, A271619, A279374, A279785, A306319, A336342, A356932.

Programs

  • Mathematica
    twiptn[n_]:=Join@@Table[Tuples[IntegerPartitions/@ptn],{ptn,IntegerPartitions[n]}];
Table[Length[Select[twiptn[n],OddQ[Length[#]]&]],{n,0,10}]
  • PARI
    R(u,y) = {1/prod(k=1, #u, 1 - u[k]*y*x^k + O(x*x^#u))}
seq(n) = {my(u=vector(n,k,numbpart(k))); Vec(R(u, 1) - R(u, -1), -(n+1))/2} \\ Andrew Howroyd, Dec 30 2022

Formula

G.f.: ((1/Product_{k>=1} (1-A000041(k)*x^k)) - (1/Product_{k>=1} (1+A000041(k)*x^k)))/2. - Andrew Howroyd, Dec 30 2022

Extensions

Terms a(26) and beyond from Andrew Howroyd, Dec 30 2022

A358834 Number of odd-length twice-partitions of n into odd-length partitions.

Original entry on oeis.org

0, 1, 1, 3, 3, 8, 11, 24, 35, 74, 109, 213, 336, 624, 986, 1812, 2832, 5002, 7996, 13783, 21936, 37528, 59313, 99598, 158356, 262547, 415590, 684372, 1079576, 1759984, 2779452, 4491596, 7069572, 11370357, 17841534, 28509802, 44668402, 70975399, 110907748
Offset: 0

Views

Author

Gus Wiseman, Dec 04 2022

Keywords

Comments

A twice-partition of n (A063834) is a sequence of integer partitions, one of each part of an integer partition of n.

Examples

			The a(1) = 1 through a(6) = 11 twice-partitions:
  (1)  (2)  (3)        (4)        (5)              (6)
            (111)      (211)      (221)            (222)
            (1)(1)(1)  (2)(1)(1)  (311)            (321)
                                  (11111)          (411)
                                  (2)(2)(1)        (21111)
                                  (3)(1)(1)        (2)(2)(2)
                                  (111)(1)(1)      (3)(2)(1)
                                  (1)(1)(1)(1)(1)  (4)(1)(1)
                                                   (111)(2)(1)
                                                   (211)(1)(1)
                                                   (2)(1)(1)(1)(1)

Links

Crossrefs

The version for set partitions is A003712.
If the parts are also odd we get A279374.
The version for multiset partitions of integer partitions is the odd-length case of A356932, ranked by A026424 /\ A356935.
This is the odd-length case of A358334.
This is the odd-lengths case of A358824.
For odd sums instead of lengths we have A358826.
The case of odd sums also is the bisection of A358827.
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, A270995, A271619, A279785, A358823, A358837.

Programs

  • Mathematica
    twiptn[n_]:=Join@@Table[Tuples[IntegerPartitions/@ptn],{ptn,IntegerPartitions[n]}];
Table[Length[Select[twiptn[n],OddQ[Length[#]]&&OddQ[Times@@Length/@#]&]],{n,0,10}]
  • PARI
    P(n,y) = {1/prod(k=1, n, 1 - y*x^k + O(x*x^n))}
R(u,y) = {1/prod(k=1, #u, 1 - u[k]*y*x^k + O(x*x^#u))}
seq(n) = {my(u=Vec(P(n,1)-P(n,-1))/2); Vec(R(u, 1) - R(u, -1), -(n+1))/2} \\ Andrew Howroyd, Dec 30 2022

Formula

G.f.: ((1/Product_{k>=1} (1-A027193(k)*x^k)) - (1/Product_{k>=1} (1+A027193(k)*x^k)))/2. - Andrew Howroyd, Dec 30 2022

Extensions

Terms a(21) and beyond from Andrew Howroyd, Dec 30 2022

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

Original entry on oeis.org

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

Views

Author

Gus Wiseman, Dec 03 2022

Keywords

Examples

			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)

Crossrefs

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.

Programs

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

Formula

G.f.: Product_{k odd} 1/(1-A000041(k)*x^k).
Showing 1-3 of 3 results.

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