A365613 a(n) = number of partitions p of n such that the greatest multiplicity of the parts of p is a part of p.

0, 1, 0, 1, 3, 2, 4, 4, 9, 11, 18, 19, 30, 36, 51, 64, 90, 107, 150, 182, 239, 294, 385, 466, 602, 733, 928, 1129, 1420, 1714, 2137, 2578, 3177, 3826, 4690, 5617, 6845, 8181, 9898, 11803, 14211, 16878, 20234, 23974, 28596, 33795, 40161, 47311, 56025, 65845

Offset

0,5

Links

Table of n, a(n) for n=0..49.

Formula

a(n) = A000041(n) - A365616(n).

Example

The partitions of 5 are [5], [4,1], [3,2], [3,1,1], [2,2,1], [2,1,1,1], [1,1,1,1,1], having greatest multiplicities 1,1,1,2,2,3,5, respectively. The partitions that include greatest multiplicity as a part are [4,1] and [2,2,1], so that a(5) = 2.

Mathematica

z = 40; f[n_] := f[n] = IntegerPartitions[n];
m[p_] := Max[Map[Length, Split[p]]]
Table[Count[f[n], p_ /; MemberQ[p, m[p]]], {n, 0, z}]

Crossrefs

Cf. A000041, A365614, A365615, A365616.

Sequence in context: A370806 A240538 A326923 * A338315 A240541 A073369

Adjacent sequences: A365610 A365611 A365612 * A365614 A365615 A365616

Keyword

nonn

Author

Clark Kimberling, Sep 17 2023

Status

approved