A392919 Number of partitions p of n such that (maximal multiplicity of the parts of p) <= (minimal part of p).

1, 1, 2, 3, 3, 5, 6, 9, 10, 15, 16, 23, 27, 35, 43, 54, 63, 82, 95, 117, 143, 172, 203, 249, 292, 350, 414, 495, 578, 688, 804, 951, 1109, 1301, 1512, 1775, 2054, 2394, 2768, 3220, 3708, 4297, 4942, 5707, 6551, 7541, 8632, 9917, 11328, 12979, 14800, 16915, 19247, 21949, 24939

Offset

1,3

Links

Table of n, a(n) for n=1..55.

Formula

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

Example

a(8) counts these 9 partitions: 8, 71, 62, 53, 521, 44, 431, 422, 332.

Prog

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

Crossrefs

Cf. A393320.

Sequence in context: A300446 A039876 A239312 * A317167 A070830 A039862

Adjacent sequences: A392916 A392917 A392918 * A392920 A392921 A392922

Keyword

nonn

Author

Seiichi Manyama, Mar 16 2026

Status

approved