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.

A289501 Number of enriched p-trees of weight n.

Original entry on oeis.org

1, 1, 2, 4, 12, 32, 112, 352, 1296, 4448, 16640, 59968, 231168, 856960, 3334400, 12679424, 49991424, 192890880, 767229952, 2998427648, 12015527936, 47438950400, 191117033472, 760625733632, 3082675150848, 12346305839104, 50223511928832, 202359539335168
Offset: 0

Views

Author

Gus Wiseman, Jul 07 2017

Keywords

Comments

An enriched p-tree of weight n is either (case 1) the number n itself, or (case 2) a sequence of two or more enriched p-trees, having a weakly decreasing sequence of weights summing to n.

Examples

			The a(4) = 12 enriched p-trees are:
  4,
  (31), ((21)1), (((11)1)1), ((111)1),
  (22), (2(11)), ((11)2), ((11)(11)),
  (211), ((11)11),
  (1111).

Links

Crossrefs

Cf. A052337, A063834, A093637, A196545, A273873, A281145, A300660.

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(n=0, 1,
      `if`(i<1, 0, b(n, i-1)+a(i)*b(n-i, min(n-i, i))))
    end:
a:= n-> `if`(n=0, 1, 1+b(n, n-1)):
seq(a(n), n=0..30);  # Alois P. Heinz, Jul 07 2017
  • Mathematica
    a[n_]:=a[n]=1+Sum[Times@@a/@y,{y,Rest[IntegerPartitions[n]]}];
Array[a,20]
(* Second program: *)
b[n_, i_] := b[n, i] = If[n == 0, 1,
     If[i<1, 0, b[n, i-1] + a[i] b[n-i, Min[n-i, i]]]];
a[n_] := If[n == 0, 1, 1 + b[n, n-1]];
a /@ Range[0, 30] (* Jean-François Alcover, May 09 2021, after Alois P. Heinz *)
  • PARI
    seq(n)={my(v=vector(n)); for(n=1, n, v[n] = 1 + polcoef(1/prod(k=1, n-1, 1 - v[k]*x^k + O(x*x^n)), n)); concat([1], v)} \\ Andrew Howroyd, Aug 26 2018

Formula

O.g.f.: (1/(1-x) + Product_{i>0} 1/(1-a(i)*x^i))/2.

A387178 Number of integer partitions of n whose parts have choosable sets of strict integer partitions.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 5, 6, 8, 10, 13, 17, 21, 27, 34, 42, 53, 65, 80, 98, 119, 146, 177, 213, 258, 309, 370, 443, 528, 628, 745, 882, 1043, 1229, 1447, 1700, 1993, 2333, 2727, 3182, 3707, 4311, 5008, 5808, 6727, 7782, 8990, 10371, 11952, 13756, 15815, 18161
Offset: 0

Views

Author

Gus Wiseman, Sep 02 2025

Keywords

Comments

First differs from A052337 in having 745 instead of 746.
We say that a sequence of nonempty sets is choosable iff it is possible to choose a different element from each set. For example, ({1,2},{1},{1,3}) is choosable because we have the choice (2,1,3), but ({1},{2},{1,3},{2,3}) is not.
a(n) is the number of integer partitions of n such that it is possible to choose a sequence of distinct strict integer partitions of each part.
Also the number of integer partitions of n with no part k whose multiplicity exceeds A000009(k).

Examples

			The partition y = (3,3,2) has sets of strict integer partitions ({(2,1),(3)},{(2,1),(3)},{(2)}), and we have the choice ((2,1),(3),(2)) or ((3),(2,1),(2)), so y is counted under a(8).
The a(1) = 1 through a(9) = 10 partitions:
  (1)  (2)  (3)    (4)    (5)    (6)      (7)      (8)      (9)
            (2,1)  (3,1)  (3,2)  (3,3)    (4,3)    (4,4)    (5,4)
                          (4,1)  (4,2)    (5,2)    (5,3)    (6,3)
                                 (5,1)    (6,1)    (6,2)    (7,2)
                                 (3,2,1)  (3,3,1)  (7,1)    (8,1)
                                          (4,2,1)  (3,3,2)  (4,3,2)
                                                   (4,3,1)  (4,4,1)
                                                   (5,2,1)  (5,3,1)
                                                            (6,2,1)
                                                            (3,3,2,1)

Crossrefs

For initial intervals instead of strict partitions we have A238873, ranks A387112.
For divisors instead of strict partitions we have A239312, ranks A368110.
The complement for divisors is A370320, ranks A355740.
For prime factors instead of strict partitions we have A370592, ranks A368100.
The complement for prime factors is A370593, ranks A355529.
The complement for initial intervals is A387118, ranks A387113.
The complement for all partitions is A387134, ranks A387577.
The complement is counted by A387137, ranks A387176.
These partitions are ranked by A387177.
For all partitions instead of just strict partitions we have A387328, ranks A387576.
The complement for constant partitions is A387329, ranks A387180.
For constant partitions instead of strict partitions we have A387330, ranks A387181.
A000041 counts integer partitions, strict A000009.
A358914 counts twice-partitions into distinct strict partitions.
A367902 counts choosable set-systems, complement A367903.
Cf. A005703, A052335, A261049, A270995, A276078, A335448, A367867, A367901, A387115.

Programs

  • Mathematica
    strptns[n_]:=Select[IntegerPartitions[n],UnsameQ@@#&];
Table[Length[Select[IntegerPartitions[n],Select[Tuples[strptns/@#],UnsameQ@@#&]!={}&]],{n,0,15}]

A387328 Number of integer partitions of n whose parts have choosable sets of integer partitions.

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 5, 7, 10, 13, 17, 22, 28, 36, 46, 58, 73, 91, 114, 141, 174, 214, 262, 320, 389, 472, 571, 688, 828, 993, 1189, 1419, 1690, 2009, 2383, 2821, 3334, 3931, 4628, 5439, 6381, 7474, 8741, 10207, 11902, 13858, 16114, 18710, 21698, 25130, 29070
Offset: 0

Views

Author

Gus Wiseman, Sep 01 2025

Keywords

Comments

First differs from A052335 at A052335(20) = 173, a(20) = 174, corresponding to the partition (4,4,4,4,4).
a(n) is the number of integer partitions of n such that it is possible to choose a sequence of distinct integer partitions, one of each part.
Also the number of integer partitions y of n with no part k whose multiplicity in y exceeds A000041(k).

Examples

			The a(1) = 1 through a(9) = 13 partitions:
  (1)  (2)  (3)   (4)   (5)    (6)    (7)    (8)     (9)
            (21)  (22)  (32)   (33)   (43)   (44)    (54)
                  (31)  (41)   (42)   (52)   (53)    (63)
                        (221)  (51)   (61)   (62)    (72)
                               (321)  (322)  (71)    (81)
                                      (331)  (332)   (333)
                                      (421)  (422)   (432)
                                             (431)   (441)
                                             (521)   (522)
                                             (3221)  (531)
                                                     (621)
                                                     (3321)
                                                     (4221)

Crossrefs

The strict case is A000009.
For initial intervals instead of partitions we have A238873, complement A387118.
For divisors instead of partitions we have A239312, complement A370320.
For prime factors instead of partitions we have A370592, ranks A368100.
The complement for prime factors is A370593, ranks A355529.
The complement is counted by A387134, ranks A387577.
For sets of strict partitions we have A387178, complement A387137.
These partitions are ranked by A387576.
A000005 counts divisors.
A000041 counts integer partitions.
Cf. A052335, A052337, A335433, A367771, A370585, A370804.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],Select[Tuples[IntegerPartitions/@#],UnsameQ@@#&]!={}&]],{n,0,15}]
Showing 1-3 of 3 results.

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