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.

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

0,12

Links

Alois P. Heinz, Rows n = 0..400, flattened

Example

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;

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 *)

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).

Sequence in context: A374212 A303903 A348515 * A170984 A114021 A239287

Adjacent sequences: A209315 A209316 A209317 * A209319 A209320 A209321

Keyword

nonn,tabf,look

Author

Alois P. Heinz, Jan 19 2013

Status

approved