A239948 Number of partitions of n such that (number of distinct parts) < least part.

1, 0, 1, 1, 2, 1, 3, 2, 4, 4, 6, 6, 9, 9, 12, 14, 17, 18, 25, 26, 32, 38, 43, 49, 62, 65, 78, 92, 103, 114, 142, 151, 175, 203, 229, 252, 302, 323, 378, 422, 477, 524, 619, 661, 758, 847, 958, 1038, 1204, 1297, 1485, 1626, 1829, 1989, 2285, 2459, 2770, 3035

Offset

0,5

Links

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

Formula

a(n) + A239952(n) = A000041(n) for n >= 0.

Example

a(10) counts these 6 partitions:  [10], [7,3], [6,4], [5,5], [4,3,3], [2,2,2,2,2].

Maple

b:= proc(n, i, d) option remember; `if`(n=0, 1, `if`(i<=d+1, 0,
      add(b(n-i*j, i-1, d+`if`(j=0, 0, 1)), j=0..n/i)))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..80);  # Alois P. Heinz, Apr 02 2014

Mathematica

z = 50; d[p_] := d[p] = Length[DeleteDuplicates[p]]; f[n_] := f[n] = IntegerPartitions[n];
Table[Count[f[n], p_ /; d[p] < Min[p]], {n, 0, z}]  (*A239948*)
Table[Count[f[n], p_ /; d[p] <= Min[p]], {n, 0, z}] (*A239949*)
Table[Count[f[n], p_ /; d[p] == Min[p]], {n, 0, z}] (*A239950*)
Table[Count[f[n], p_ /; d[p] > Min[p]], {n, 0, z}]  (*A239951*)
Table[Count[f[n], p_ /; d[p] >= Min[p]], {n, 0, z}] (*A239952*)
b[n_, i_, d_] := b[n, i, d] = If[n==0, 1, If[i <= d+1, 0, Sum[b[n-i*j, i-1, d + If[j==0, 0, 1]], {j, 0, n/i}]]]; a[n_] := b[n, n, 0]; Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Oct 12 2015, after Alois P. Heinz *)

Crossrefs

Cf. A239949, A239950, A239951, A239952.

Sequence in context: A001687 A159072 A116928 * A034391 A239243 A206738

Adjacent sequences: A239945 A239946 A239947 * A239949 A239950 A239951

Keyword

nonn,easy

Author

Clark Kimberling, Mar 30 2014

Status

approved