A182485 Number of partitions of n into exactly k different parts with distinct multiplicities; triangle T(n,k), n>=0, 0<=k<=max{i:A000292(i)<=n}, read by rows.

1, 0, 1, 0, 2, 0, 2, 0, 3, 1, 0, 2, 3, 0, 4, 3, 0, 2, 8, 0, 4, 9, 0, 3, 12, 0, 4, 16, 1, 0, 2, 22, 4, 0, 6, 20, 5, 0, 2, 31, 12, 0, 4, 35, 16, 0, 4, 34, 24, 0, 5, 44, 33, 0, 2, 51, 52, 0, 6, 53, 57, 0, 2, 62, 89, 0, 6, 65, 100, 1, 0, 4, 68, 131, 5, 0, 4, 87

Offset

0,5

Links

Alois P. Heinz, Rows n = 0..120, flattened

Example

T(0,0) = 1: [].
T(1,1) = 1: [1].
T(2,1) = 2: [1,1], [2].
T(4,1) = 3: [1,1,1,1], [2,2], [4].
T(4,2) = 1: [2,1,1]; part 2 occurs once and part 1 occurs twice.
T(5,2) = 3: [2,1,1,1], [2,2,1], [3,1,1].
T(7,2) = 8: [2,1,1,1,1,1], [2,2,1,1,1], [2,2,2,1], [3,1,1,1,1], [3,2,2], [3,3,1], [4,1,1,1], [5,1,1].
T(10,1) = 4: [1,1,1,1,1,1,1,1,1,1], [2,2,2,2,2], [5,5], [10].
T(10,3) = 1: [3,2,2,1,1,1].
Triangle T(n,k) begins:
  1;
  0, 1;
  0, 2;
  0, 2;
  0, 3,  1;
  0, 2,  3;
  0, 4,  3;
  0, 2,  8;
  0, 4,  9;
  0, 3, 12;
  0, 4, 16, 1;

Maple

b:= proc(n, i, t, s) option remember;
      `if`(nops(s)>t, 0, `if`(n=0, 1, `if`(i<1, 0, b(n, i-1, t, s)+
      add(`if`(j in s, 0, b(n-i*j, i-1, t, s union {j})), j=1..n/i))))
    end:
g:= proc(n) local i; for i while i*(i+1)*(i+2)/6<=n do od; i-1 end:
T:= n-> seq(b(n, n, k, {}) -b(n, n, k-1, {}), k=0..g(n)):
seq(T(n), n=0..30);

Mathematica

b[n_, i_, t_, s_] := b[n, i, t, s] = If[Length[s] > t, 0, If[n == 0, 1, If[i < 1, 0, b[n, i-1, t, s] + Sum[If[MemberQ[s, j], 0, b[n-i*j, i-1, t, s ~Union~ {j}]], {j, 1, n/i}]]]]; g[n_] := Module[{i}, For[ i = 1, i*(i+1)*(i+2)/6 <= n , i++]; i-1 ]; t[n_] := Table [b[n, n, k, {}] - b[n, n, k-1, {}], {k, 0, g[n]}]; Table [t[n], {n, 0, 30}] // Flatten (* Jean-François Alcover, Dec 19 2013, translated from Maple *)

Crossrefs

Columns k=0-7 give: A000007, A000005, A182473, A212114, A212115, A212116, A212117, A212118.

Row sums give: A098859.

First row with length (t+1): A000292(t).

Cf. A242896 (the same for compositions):

Sequence in context: A094098 A326844 A261079 * A164960 A124137 A164092

Adjacent sequences: A182482 A182483 A182484 * A182486 A182487 A182488

Keyword

nonn,tabf

Author

Alois P. Heinz, May 01 2012

Status

approved