A394459 Number of partitions p of n such that 3 * (the greatest multiplicity of the parts of p) is a part of p.

0, 0, 1, 1, 1, 1, 1, 3, 3, 6, 6, 10, 11, 17, 22, 30, 35, 47, 57, 77, 93, 121, 146, 190, 228, 290, 351, 441, 531, 661, 795, 981, 1174, 1440, 1722, 2096, 2497, 3021, 3594, 4324, 5124, 6137, 7256, 8652, 10204, 12115, 14252, 16863, 19784, 23325, 27305, 32086, 37470, 43904, 51161, 59771

Offset

1,8

Links

Table of n, a(n) for n=1..56.

Formula

G.f.: Sum_{j>=1} Product_{k>=1} (-delta(3*j,k) + (1-q^((j+1)*k))/(1-q^k)) - Product_{k>=1} (-delta(3*j,k) + (1-q^(j*k))/(1-q^k)), where delta(j,k) is the Kronecker delta.

Example

a(10) counts these 6 partitions: 73, 631, 622, 6211, 532, 4321.

Prog

(PARI) my(N=60, q='q+O('q^N)); concat([0, 0], Vec(sum(j=1, N, prod(k=1, 3*N, -(3*j==k)+(1-q^((j+1)*k))/(1-q^k))-prod(k=1, 3*N, -(3*j==k)+(1-q^(j*k))/(1-q^k)))))

Crossrefs

Cf. A365613, A394458.

Sequence in context: A343481 A237665 A355394 * A088528 A363241 A220153

Adjacent sequences: A394456 A394457 A394458 * A394460 A394461 A394462

Keyword

nonn

Author

Seiichi Manyama, Mar 21 2026

Status

approved