A394323 Number of partitions p of n such that (minimal multiplicity of the parts of p) >= 2 * (minimal part of p).

0, 1, 1, 1, 1, 2, 2, 5, 4, 7, 7, 11, 10, 17, 17, 24, 26, 38, 38, 55, 58, 77, 85, 113, 118, 159, 171, 219, 239, 306, 327, 421, 454, 564, 620, 769, 829, 1031, 1122, 1366, 1496, 1816, 1977, 2395, 2616, 3122, 3433, 4090, 4461, 5309, 5811, 6840, 7515, 8819, 9649, 11311, 12390, 14410

Offset

1,6

Links

Table of n, a(n) for n=1..58.

Formula

G.f.: Sum_{j>=1} q^(2*j^2)/(1-q^j) * Product_{k>=j+1} (1+q^(2*j*k)/(1-q^k)).

Example

a(8) counts these 5 partitions: 3311, 2222, 22211, 221111, 11111111.

Prog

(PARI) my(N=60, q='q+O('q^N)); concat(0, Vec(sum(j=1, N, q^(2*j^2)/(1-q^j)*prod(k=j+1, N, 1+q^(2*j*k)/(1-q^k)))))

Crossrefs

Cf. A394324.

Sequence in context: A162200 A290289 A000019 * A318196 A367213 A081177

Adjacent sequences: A394320 A394321 A394322 * A394324 A394325 A394326

Keyword

nonn

Author

Seiichi Manyama, Mar 16 2026

Status

approved