A239953 Number of partitions of n such that twice the least part is the number of distinct parts.

0, 0, 0, 1, 2, 4, 5, 8, 9, 12, 14, 17, 18, 23, 25, 28, 33, 39, 44, 54, 61, 77, 92, 112, 131, 167, 194, 246, 280, 352, 401, 501, 562, 697, 779, 939, 1055, 1274, 1401, 1684, 1846, 2186, 2408, 2825, 3103, 3617, 3969, 4583, 5045, 5801, 6367, 7304, 8050, 9150

Offset

0,5

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Example

a(6) counts these 5 partitions :  51, 42, 3111, 22111, 21111.

Maple

b:= proc(n, i, d) option remember; `if`(2*min(i, n)<d+1, 0,
      `if`(irem(n, i)=0 and 2*i=d+1, 1, b(n, i-1, d)+
      add(b(n-i*j, i-1, d+1), j=1..n/i)))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..60);  # Alois P. Heinz, Apr 02 2014

Mathematica

z = 60; d[p_] := d[p] = Length[DeleteDuplicates[p]]; Table[Count[ IntegerPartitions[n], p_ /; d[p] == 2 Min[p]], {n, 0, z}]  (* A239953 *)
(* Second program: *)
b[n_, i_, d_] := b[n, i, d] = If[2*Min[i, n] < d + 1, 0,
     If[Mod[n, i] == 0 && 2*i == d + 1, 1, b[n, i - 1, d] +
     Sum[b[n - i*j, i - 1, d + 1], {j, 1, n/i}]]];
a[n_] := b[n, n, 0];
a /@ Range[0, 60] (* Jean-François Alcover, May 31 2021, after Alois P. Heinz *)

Crossrefs

Cf. A239948.

Sequence in context: A348458 A339816 A002541 * A321324 A343013 A259558

Adjacent sequences: A239950 A239951 A239952 * A239954 A239955 A239956

Keyword

nonn,easy

Author

Clark Kimberling, Mar 30 2014

Status

approved