A394251 Number of partitions p of n such that (maximal multiplicity of the parts of p) <= 3 * (number of distinct parts of p).

1, 1, 2, 3, 4, 6, 10, 14, 20, 28, 38, 51, 69, 92, 124, 160, 211, 271, 352, 447, 571, 725, 917, 1151, 1441, 1797, 2233, 2767, 3407, 4193, 5140, 6289, 7662, 9323, 11305, 13695, 16527, 19916, 23929, 28717, 34377, 41079, 48980, 58303, 69273, 82186, 97330, 115080

Offset

0,3

Links

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

Formula

G.f.: Sum_{i>=0} [z^i] Product_{j>=1} (1 + z * Sum_{k=1..3*i} q^(j*k)).

Example

a(6) counts these 10 partitions: 6, 51, 42, 411, 33, 321, 3111, 222, 2211, 21111.

Prog

(PARI) my(N=50, q='q+O('q^N)); Vec(sum(i=0, N, polcoef(prod(j=1, N, 1+z*sum(k=1, 3*i, q^(j*k))), i, z)))

Crossrefs

Cf. A000041, A240306, A394250.

Sequence in context: A200928 A318558 A347867 * A391620 A256248 A089223

Adjacent sequences: A394248 A394249 A394250 * A394252 A394253 A394254

Keyword

nonn

Author

Seiichi Manyama, Mar 13 2026

Status

approved