A394460 Number of partitions p of n such that 2 * (the least multiplicity of the parts of p) is a part of p.

0, 1, 1, 1, 3, 3, 6, 9, 13, 18, 28, 36, 51, 70, 93, 121, 166, 213, 279, 359, 465, 589, 759, 948, 1202, 1506, 1886, 2334, 2915, 3584, 4426, 5428, 6661, 8106, 9903, 11992, 14552, 17570, 21210, 25460, 30629, 36634, 43840, 52269, 62311, 73990, 87918, 104057, 123163, 145375, 171517

Offset

1,5

Links

Table of n, a(n) for n=1..51.

Formula

G.f.: Sum_{j>=1} Product_{k>=1} (1 - delta(2*j,k) + q^(j*k)/(1-q^k)) - Product_{k>=1} (1 - delta(2*j,k) + q^((j+1)*k)/(1-q^k)), where delta(j,k) is the Kronecker delta.

Example

a(8) counts these 9 partitions: 62, 521, 44, 422, 4211, 332, 3221, 32111, 2111111.

Prog

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

Crossrefs

Cf. A365614, A394461.

Sequence in context: A323451 A280240 A058628 * A035528 A341241 A300300

Adjacent sequences: A394457 A394458 A394459 * A394461 A394462 A394463

Keyword

nonn

Author

Seiichi Manyama, Mar 21 2026

Status

approved