A244515 Number of partitions of n where the minimal multiplicity of any part is 2.

0, 1, 0, 1, 0, 2, 1, 4, 2, 6, 4, 9, 6, 16, 9, 23, 18, 34, 27, 51, 40, 75, 63, 103, 90, 152, 130, 208, 191, 286, 267, 402, 368, 546, 518, 730, 709, 998, 954, 1322, 1305, 1751, 1740, 2330, 2299, 3056, 3074, 3968, 4031, 5202, 5249, 6721, 6877, 8642, 8888, 11147, 11432, 14248, 14747, 18097, 18838, 23093, 23938, 29186, 30489

Offset: 1

Author

Joerg Arndt and Alois P. Heinz, Jun 29 2014

Keywords

• nonn

Examples

			From _Gus Wiseman_, Jul 03 2019: (Start)
The a(2) = 1 through a(12) = 9 partitions are the following (empty columns not shown). The Heinz numbers of these partitions are given by A325240.
  11  22  33    22111  44      33111    55        33311      66
          2211         3311    2211111  3322      44111      4422
                       22211            4411      3311111    5511
                       221111           222211    221111111  33222
                                        331111               332211
                                        22111111             441111
                                                             2222211
                                                             33111111
                                                             2211111111
(End)
		

Links

• Joerg Arndt and Alois P. Heinz, Table of n, a(n) for n = 1..1000

Crossrefs

Column k = 2 of A243978.

Cf. A000041, A007690, A008284, A116608, A325240, A325242.

Programs

Maple

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1, k) +add(b(n-i*j, i-1, k), j=max(1, k)..n/i)))
    end:
a:= n-> b(n$2, 2) -b(n$2, 3):
seq(a(n), n=1..80);

Mathematica

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, b[n, i - 1, k] + Sum[b[n - i*j, i - 1, k], {j, Max[1, k], n/i}]]];
a[n_] := b[n, n, 2] - b[n, n, 3];
Array[a, 80] (* Jean-François Alcover, May 01 2018, translated from Maple *)
Table[Length[Select[IntegerPartitions[n],Min@@Length/@Split[#]==2&]],{n,0,30}] (* Gus Wiseman, Jul 03 2019 *)