A239961 Number of partitions of n such that (number of distinct parts) = number of 2's.

1, 0, 1, 0, 0, 1, 1, 2, 2, 2, 4, 4, 6, 9, 10, 12, 19, 21, 24, 36, 44, 49, 66, 81, 100, 123, 144, 180, 229, 265, 317, 391, 473, 566, 675, 798, 968, 1154, 1354, 1621, 1926, 2241, 2675, 3170, 3691, 4345, 5113, 5956, 7002, 8182, 9503, 11095, 12919, 14976, 17446

Offset

0,8

Links

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

Example

a(10) counts these 4 partitions :  622, 3322, 32221, 22111111.

Maple

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1)+`if`(i=2, 0, add(b(n-2-i*j, i-1), j=1..(n-2)/i))))
    end:
a:= n-> `if`(n=0, 1, b(n-2$2)):
seq(a(n), n=0..70);  # Alois P. Heinz, Apr 03 2014

Mathematica

z = 54; d[p_] := d[p] = Length[DeleteDuplicates[p]]; Table[Count[IntegerPartitions[n], p_ /; d[p] == Count[p, 2]], {n, 0, z}]
(* Second program: *)
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, b[n, i-1] +
     If[i == 2, 0, Sum[b[n-2-i*j, i-1], {j, 1, (n-2)/i}]]]];
a[n_] := If[n == 0, 1, b[n-2, n-2]];
a /@ Range[0, 70] (* Jean-François Alcover, Jun 05 2021, after Alois P. Heinz *)

Crossrefs

Cf. A239960.

Sequence in context: A326458 A326544 A326683 * A365541 A308855 A301588

Adjacent sequences: A239958 A239959 A239960 * A239962 A239963 A239964

Keyword

nonn,easy

Author

Clark Kimberling, Mar 30 2014

Status

approved