A393320 Number of partitions p of n such that (maximal multiplicity of the parts of p) <= 2 * (minimal part of p).

1, 2, 2, 4, 5, 8, 9, 14, 18, 23, 30, 41, 49, 65, 81, 103, 126, 161, 194, 244, 295, 364, 440, 539, 646, 785, 937, 1131, 1345, 1614, 1908, 2278, 2688, 3187, 3749, 4427, 5186, 6100, 7127, 8348, 9722, 11351, 13180, 15338, 17765, 20604, 23808, 27538, 31735, 36611, 42100, 48439, 55581

Offset

1,2

Links

Table of n, a(n) for n=1..53.

Formula

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

Example

a(8) counts these 14 partitions: 8, 71, 62, 611, 53, 521, 44, 431, 422, 4211, 332, 3311, 3221, 2222.

Prog

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

Crossrefs

Cf. A000041, A394321.

Sequence in context: A304332 A183564 A222707 * A326525 A326630 A317810

Adjacent sequences: A393317 A393318 A393319 * A393321 A393322 A393323

Keyword

nonn

Author

Seiichi Manyama, Mar 16 2026

Status

approved