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.

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A209318 Number T(n,k) of partitions of n with k parts in which no part occurs more than twice; triangle T(n,k), n>=0, 0<=k<=A055086(n), read by rows.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 2, 2, 0, 1, 3, 2, 1, 0, 1, 3, 4, 1, 0, 1, 4, 5, 3, 0, 1, 4, 6, 4, 1, 0, 1, 5, 8, 6, 2, 0, 1, 5, 10, 8, 3, 0, 1, 6, 11, 12, 5, 1, 0, 1, 6, 14, 14, 8, 1, 0, 1, 7, 16, 19, 11, 3, 0, 1, 7, 18, 23, 16, 5
Offset: 0

Views

Author

Alois P. Heinz, Jan 19 2013

Keywords

Examples

			T(8,3) = 5: [6,1,1], [5,2,1], [4,3,1], [4,2,2], [3,3,2].
T(8,4) = 3: [4,2,1,1], [3,3,1,1], [3,2,2,1].
T(9,3) = 6: [7,1,1], [6,2,1], [5,3,1], [4,4,1], [5,2,2], [4,3,2].
T(9,4) = 4: [5,2,1,1], [4,3,1,1], [4,2,2,1], [3,3,2,1].
T(9,5) = 1: [3,2,2,1,1].
Triangle begins:
  1;
  0, 1;
  0, 1, 1;
  0, 1, 1;
  0, 1, 2, 1;
  0, 1, 2, 2;
  0, 1, 3, 2, 1;
  0, 1, 3, 4, 1;
  0, 1, 4, 5, 3;
  0, 1, 4, 6, 4, 1;

Links

Crossrefs

Columns k=0-10 give: A000007, A057427, A004526, A230059 (conjectured), A320592, A320593, A320594, A320595, A320596, A320597, A320598.
Row sums give: A000726.
Row lengths give: A000267.
Cf. A002620, A008289 (no part more than once), A055086, A117147 (no part more than 3 times).

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(expand(b(n-i*j, i-1)*x^j), j=0..min(2, n/i))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2)):
seq(T(n), n=0..20);
  • Mathematica
    max = 15; g = -1+Product[1+t*x^j+t^2*x^(2j), {j, 1, max}]; t[n_, k_] := SeriesCoefficient[g, {x, 0, n}, {t, 0, k}]; t[0, 0] = 1; Table[Table[t[n, k], {k, 0, n}] /. {a__, 0 ..} -> {a}, {n, 0, max}] // Flatten (* Jean-François Alcover, Jan 08 2014 *)

A236293 Triangular array T: T(n,1) = number of constant partitions of n; T(n,k) = number of nonconstant partitions of n that have length k, for k = 2..n-1, for n >= 2.

Original entry on oeis.org

1, 2, 2, 1, 3, 1, 1, 2, 2, 2, 1, 4, 2, 2, 2, 1, 2, 3, 4, 3, 2, 1, 4, 3, 5, 4, 3, 2, 1, 3, 4, 6, 6, 5, 3, 2, 1, 4, 4, 8, 9, 6, 5, 3, 2, 1, 2, 5, 10, 11, 10, 7, 5, 3, 2, 1, 6, 5, 11, 14, 13, 10, 7, 5, 3, 2, 1, 2, 6, 14, 18, 18, 14, 11, 7, 5, 3, 2, 1, 4, 6, 16
Offset: 1

Views

Author

Clark Kimberling, Jan 22 2014

Keywords

Comments

This array occurs naturally in a method for counting the distinct cyclic permutations of the partitions of n (A236292). The row sums, limit of column n, and limit of reversed row n are given by A000041, and (column 1) = A000005. Does column 3 give the nonzero terms of A230059?

Examples

			First ten rows:
1
2
2 .. 1
3 .. 1 .. 1
2 .. 2 .. 2 .. 1
4 .. 2 .. 2 .. 2 .. 1
2 .. 3 .. 4 .. 3 .. 2 .. 1
4 .. 3 .. 5 .. 4 .. 3 .. 2 .. 1
3 .. 4 .. 6 .. 6 .. 5 .. 3 .. 2 .. 1
4 .. 4 .. 8 .. 9 .. 6 .. 5 .. 3 .. 2 .. 1
(row 6) = (4,2,2,2,1).  The 5 numbers in (4,2,2,2,1) count these partitions:  (6, 33, 222, 1111);  (51, 42); (411, 321); (3111, 2211);  (211111).  ("Constant partition" is exemplified by 6, 33, 222, 1111; i.e., all the parts are the same number.)

Links

Crossrefs

Cf. A000005, A236292, A230059.

Programs

  • Mathematica
    t = Map[Flatten[{Length[#[[1]]], Transpose[Tally[Map[Length, #[[2]]]]][[2]]} &[GatherBy[IntegerPartitions[#], Length[Union[#]] > 1 &]]] &, Range[3, 20]] u = Flatten[t]; Flatten[Prepend[u, {1, 2}]]
(* Peter J. C. Moses, Jan 21 2014 *)

Formula

Row n: (d(n), f(2), f(3),..., f(n-1)), where d(n) = (number of divisors of n) = (number of constant partitions of n), and f(k) = number of nonconstant partitions of n, for k = 2,3,...,n-1.

A340445 Number of partitions of n into 3 parts that are not all the same.

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 2, 4, 5, 6, 8, 10, 11, 14, 16, 18, 21, 24, 26, 30, 33, 36, 40, 44, 47, 52, 56, 60, 65, 70, 74, 80, 85, 90, 96, 102, 107, 114, 120, 126, 133, 140, 146, 154, 161, 168, 176, 184, 191, 200, 208, 216, 225, 234, 242, 252, 261, 270, 280, 290, 299, 310, 320, 330
Offset: 0

Views

Author

Wesley Ivan Hurt, Jan 07 2021

Keywords

Comments

Conjecturally the same as A230059 (apart from the offset). - R. J. Mathar, Jan 14 2021

Examples

			a(6) = 2; [4,1,1], [3,2,1] ( [2,2,2] not counted ),
a(7) = 4; [5,1,1], [4,2,1], [3,3,1], [3,2,2],
a(8) = 5; [6,1,1], [5,2,1], [4,3,1], [4,2,2], [3,3,2],
a(9) = 6; [7,1,1], [6,2,1], [5,3,1], [4,4,1], [5,2,2], [4,3,2] ( [3,3,3] not counted ).

Links

Crossrefs

Cf. A036410, A069905.

Programs

  • Mathematica
    Table[Sum[Sum[(1 - KroneckerDelta[i, k, n - i - k]), {i, k, Floor[(n - k)/2]}], {k, Floor[n/3]}], {n, 0, 80}]

Formula

a(n) = Sum_{k=1..floor(n/3)} Sum_{i=k..floor((n-k)/2)} (1 - [k = i = n-i-k]), where [ ] is the (generalized) Iverson bracket.
a(n) = Sum_{k=1..floor(n/3)} Sum_{i=k..floor((n-k)/2)} (1 - [k = i] * [2*i = n-k] * [2*k = n-i]), where [ ] is the Iverson bracket.
From Alois P. Heinz, Jan 07 2021: (Start)
G.f.: x^4*(x^2-x-1)/((x+1)*(x^2+x+1)*(x-1)^3).
a(n) = a(n-1) + a(n-2) - a(n-4) - a(n-5) + a(n-6), n>6. (End)
a(n) = A036410(n-1)-1. - Hugo Pfoertner, Jan 09 2021
a(n) + A079978(n) = A069905(n), n>0. - R. J. Mathar, Jan 18 2021
72*a(n) = -16*A099837(n+3) -9*(-1)^n +6*n^2 -31. - R. J. Mathar, Jun 09 2022
Showing 1-3 of 3 results.

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