A394189 Number of partitions of n such that (number of distinct parts) < minimal multiplicity of the parts.

1, 0, 1, 1, 2, 1, 3, 1, 3, 3, 4, 3, 8, 5, 7, 11, 11, 10, 17, 14, 19, 22, 21, 22, 35, 28, 33, 41, 45, 42, 62, 52, 68, 76, 80, 81, 117, 98, 121, 136, 156, 147, 196, 178, 224, 236, 257, 255, 340, 304, 363, 382, 440, 422, 526, 496, 599, 606, 677, 670, 845, 780, 910, 934, 1079, 1051, 1249

Offset

0,5

Links

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

Formula

G.f.: Sum_{j>=0} [z^j] Product_{k>=1} (1 + z*q^((j+1)*k)/(1-q^k)).

Example

a(12) = 8 counts these partitions: 66, 444, 3333, 333111, 222222, 22221111, 222111111, 111111111111.

Prog

(PARI) my(N=70, q='q+O('q^N)); Vec(sum(j=0, N, polcoef(prod(k=1, N, 1+z*q^((j+1)*k)/(1-q^k)), j, z)))

Crossrefs

Cf. A239966, A240305, A394188.

Sequence in context: A100714 A339364 A050123 * A081386 A213712 A143802

Adjacent sequences: A394186 A394187 A394188 * A394190 A394191 A394192

Keyword

nonn

Author

Seiichi Manyama, Mar 12 2026

Status

approved