A239952 Number of partitions of n such that (number of distinct parts) >= least part.

0, 1, 1, 2, 3, 6, 8, 13, 18, 26, 36, 50, 68, 92, 123, 162, 214, 279, 360, 464, 595, 754, 959, 1206, 1513, 1893, 2358, 2918, 3615, 4451, 5462, 6691, 8174, 9940, 12081, 14631, 17675, 21314, 25637, 30763, 36861, 44059, 52555, 62600, 74417, 88287, 104600, 123716

Offset

0,4

Links

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

Formula

a(n) + A239948(n) = A000041(n) for n >= 0.

Example

a(6) counts these 8 partitions:  51, 42, 411, 321, 3111, 2211, 21111, 111111.

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-> combinat[numbpart](n) -b(n$2, 0):
seq(a(n), n=0..60);  # 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_] := PartitionsP[n] - b[n, n, 0]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Nov 17 2015, after Alois P. Heinz *)

Crossrefs

Cf. A239948, A239949, A239950, A239951.

Sequence in context: A022943 A068491 A364796 * A353902 A240076 A266771

Adjacent sequences: A239949 A239950 A239951 * A239953 A239954 A239955

Keyword

nonn,easy

Author

Clark Kimberling, Mar 30 2014

Status

approved