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.

A182616 Number of partitions of 2n that contain odd parts.

Original entry on oeis.org

0, 1, 3, 8, 17, 35, 66, 120, 209, 355, 585, 946, 1498, 2335, 3583, 5428, 8118, 12013, 17592, 25525, 36711, 52382, 74173, 104303, 145698, 202268, 279153, 383145, 523105, 710655, 960863, 1293314, 1733281, 2313377, 3075425, 4073085, 5374806, 7067863, 9263076
Offset: 0

Views

Author

Omar E. Pol, Dec 03 2010

Keywords

Comments

Bisection (even part) of A086543.

Examples

			For n=3 the partitions of 2n are
6 ....................... does not contains odd parts
3 + 3 ................... contains odd parts ........... *
4 + 2 ................... does not contains odd parts
2 + 2 + 2 ............... does not contains odd parts
5 + 1 ................... contains odd parts ........... *
3 + 2 + 1 ............... contains odd parts ........... *
4 + 1 + 1 ............... contains odd parts ........... *
2 + 2 + 1 + 1 ........... contains odd parts ........... *
3 + 1 + 1 + 1 ........... contains odd parts ........... *
2 + 1 + 1 + 1 + 1 ....... contains odd parts ........... *
1 + 1 + 1 + 1 + 1 + 1 ... contains odd parts ........... *
There are 8 partitions of 2n that contain odd parts.
Also p(2n)-p(n) = p(6)-p(3) = 11-3 = 8, where p(n) is the number of partitions of n, so a(3)=8.
From _Gus Wiseman_, Oct 18 2023: (Start)
For n > 0, also the number of integer partitions of 2n that do not contain n, ranked by A366321. For example, the a(1) = 1 through a(4) = 17 partitions are:
  (2)  (4)     (6)       (8)
       (31)    (42)      (53)
       (1111)  (51)      (62)
               (222)     (71)
               (411)     (332)
               (2211)    (521)
               (21111)   (611)
               (111111)  (2222)
                         (3221)
                         (3311)
                         (5111)
                         (22211)
                         (32111)
                         (221111)
                         (311111)
                         (2111111)
                         (11111111)
(End)

Crossrefs

Cf. A304710.
Bisection of A086543, with ranks A366322.
The case of all odd parts is A035294, bisection of A000009.
The strict case is A365828.
These partitions have ranks A366530.
A000041 counts integer partitions, strict A000009.
A006477 counts partitions with at least one odd and even part, ranks A366532.
A047967 counts partitions with at least one even part, ranks A324929.
A086543 counts partitions of n not containing n/2, ranks A366319.
A366527 counts partitions of 2n with an even part, ranks A366529.
Cf. A001522, A006827, A058695, A078408, A079122, A231429, A365543, A366321.

Programs

  • Maple
    with(combinat): a:= n-> numbpart(2*n) -numbpart(n): seq(a(n), n=0..35);
  • Mathematica
    Table[Length[Select[IntegerPartitions[2n],n>0&&FreeQ[#,n]&]],{n,0,15}] (* Gus Wiseman, Oct 11 2023 *)
Table[Length[Select[IntegerPartitions[2n],Or@@OddQ/@#&]],{n,0,15}] (* Gus Wiseman, Oct 11 2023 *)

Formula

a(n) = A000041(2*n) - A000041(n).

Extensions

Edited by Alois P. Heinz, Dec 03 2010

A367094 Irregular triangle read by rows with trailing zeros removed where T(n,k) is the number of integer partitions of 2n whose number of submultisets summing to n is k.

Original entry on oeis.org

0, 1, 1, 1, 2, 2, 1, 5, 3, 3, 8, 4, 9, 1, 17, 6, 16, 1, 2, 24, 7, 33, 4, 9, 46, 11, 52, 3, 18, 1, 4, 64, 12, 91, 6, 38, 3, 15, 1, 1, 107, 17, 138, 9, 68, 2, 28, 2, 12, 0, 2, 147, 19, 219, 12, 117, 6, 56, 3, 34, 2, 9, 0, 3
Offset: 0

Views

Author

Gus Wiseman, Nov 07 2023

Keywords

Examples

			The partition (3,2,2,1) has two submultisets summing to 4, namely {2,2} and {1,3}, so it is counted under T(4,2).
The partition (2,2,1,1,1,1) has three submultisets summing to 4, namely {1,1,1,1}, {1,1,2}, and {2,2}, so it is counted under T(4,3).
Triangle begins:
    0   1
    1   1
    2   2   1
    5   3   3
    8   4   9   1
   17   6  16   1   2
   24   7  33   4   9
   46  11  52   3  18   1   4
   64  12  91   6  38   3  15   1   1
  107  17 138   9  68   2  28   2  12   0   2
  147  19 219  12 117   6  56   3  34   2   9   0   3
Row n = 4 counts the following partitions:
  (8)     (44)        (431)      (221111)
  (71)    (3311)      (422)
  (62)    (2222)      (4211)
  (611)   (11111111)  (41111)
  (53)                (3221)
  (521)               (32111)
  (5111)              (311111)
  (332)               (22211)
                      (2111111)

Crossrefs

Row sums w/o the first column are A002219, ranks A357976, strict A237258.
Column k = 0 is A006827.
Row sums are A058696.
Column k = 1 is A108917.
The corresponding rank statistic is A357879 (without empty rows).
A000041 counts integer partitions, strict A000009.
A182616 counts partitions of 2n that do not contain n, ranks A366321.
A182616 counts partitions of 2n with at least one odd part, ranks A366530.
A276024 counts positive subset-sums of partitions, strict A284640.
A304792 counts subset-sums of partitions, rank statistic A299701.
A365543 counts partitions of n with a submultiset summing to k.
Cf. A046663, A064914, A079122, A122768, A213074, A231429, A235130, A237194.

Programs

  • Mathematica
    t=Table[Length[Select[IntegerPartitions[2n], Count[Total/@Union[Subsets[#]],n]==k&]], {n,0,5}, {k,0,1+PartitionsP[n]}];
Table[NestWhile[Most,t[[i]],Last[#]==0&], {i,Length[t]}]

Formula

T(n,1) = A108917(n).

A366527 Number of integer partitions of 2n containing at least one even part.

Original entry on oeis.org

0, 1, 3, 7, 16, 32, 62, 113, 199, 339, 563, 913, 1453, 2271, 3496, 5308, 7959, 11798, 17309, 25151, 36225, 51748, 73359, 103254, 144363, 200568, 277007, 380437, 519715, 706412, 955587, 1286762, 1725186, 2303388, 3063159, 4058041, 5356431, 7045454, 9235841
Offset: 0

Views

Author

Gus Wiseman, Oct 16 2023

Keywords

Comments

Also partitions of 2n with even product.

Examples

			The a(1) = 1 through a(4) = 16 partitions:
  (2)  (4)    (6)      (8)
       (22)   (42)     (44)
       (211)  (222)    (62)
              (321)    (332)
              (411)    (422)
              (2211)   (431)
              (21111)  (521)
                       (611)
                       (2222)
                       (3221)
                       (4211)
                       (22211)
                       (32111)
                       (41111)
                       (221111)
                       (2111111)

Crossrefs

This is the even bisection of A047967.
For odd instead of even parts we have A182616, ranks A366321 or A366528.
These partitions have ranks A366529, subset of A324929.
A000041 counts integer partitions, strict A000009.
A006477 counts partitions w/ at least one odd and even part, ranks A366532.
A086543 counts partitions of n not containing n/2, ranks A366319.
A086543 counts partitions w/o odds, ranks A366322, even bisection A182616.
Cf. A001255, A006827, A035363, A064914, A078408, A086543, A231429, A304710, A365828.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[2n],Or@@EvenQ/@#&]],{n,0,15}]

Formula

a(n) = A000041(2n) - A000009(2n).
Showing 1-3 of 3 results.

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