A239964 Number of partitions of n such that (number of distinct parts) = maximal multiplicity of the parts.

0, 1, 1, 1, 2, 3, 3, 4, 6, 5, 11, 12, 15, 21, 26, 30, 41, 52, 56, 82, 99, 118, 144, 189, 221, 279, 327, 405, 491, 600, 699, 860, 1038, 1230, 1470, 1754, 2106, 2487, 2970, 3489, 4148, 4883, 5779, 6763, 8024, 9284, 11006, 12780, 15029, 17452, 20405, 23660

Offset

0,5

Links

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Formula

G.f.: Sum_{i>=1} [z^i] ( Product_{j>=1} (1 + z * Sum_{k=1..i} q^(j*k)) - Product_{j>=1} (1 + z * Sum_{k=1..i-1} q^(j*k)) ). - Seiichi Manyama, Mar 13 2026

Example

a(8) counts these partitions:  8, 611, 422, 332, 3311, 32111.

Mathematica

z = 58; d[p_] := d[p] = Length[DeleteDuplicates[p]]; m[p_] := Max[Map[Length, Split[p]]]; Table[Count[IntegerPartitions[n], p_ /; d[p] == m[p]], {n, 0, z}]

Crossrefs

Cf. A239948, A239966, A240305, A240306.

Sequence in context: A382079 A257241 A382460 * A290585 A106464 A093003

Adjacent sequences: A239961 A239962 A239963 * A239965 A239966 A239967

Keyword

nonn,easy

Author

Clark Kimberling, Mar 30 2014

Status

approved