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

Author

Clark Kimberling, Mar 30 2014

Keywords

• nonn
• easy

Examples

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

Links

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

Crossrefs

Cf. A239948.

Programs

Maple

b:= proc(n, i, d) option remember; `if`(2*min(i, 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 *)