A365616 a(n) = number of partitions p of n such that the greatest multiplicity of the parts of p is not a part of p.

1, 0, 2, 2, 2, 5, 7, 11, 13, 19, 24, 37, 47, 65, 84, 112, 141, 190, 235, 308, 388, 498, 617, 789, 973, 1225, 1508, 1881, 2298, 2851, 3467, 4264, 5172, 6317, 7620, 9266, 11132, 13456, 16117, 19382, 23127, 27705, 32940, 39287, 46579, 55339, 65397, 77443, 91248

Offset

0,3

Links

Table of n, a(n) for n=0..48.

Formula

a(n) = A000041(n) - A365613(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 do not include the greatest multiplicity as a part are [5], [3,2], [3,1,1], [2,1,1,1],and [1,1,1,1,1], so that a(5) = 5.

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, A365613, A365614, A365615.

Sequence in context: A145876 A240540 A039878 * A162145 A208050 A322176

Adjacent sequences: A365613 A365614 A365615 * A365617 A365618 A365619

Keyword

nonn

Author

Clark Kimberling, Sep 17 2023

Status

approved