A394024 Number of partitions p of n with multiplicity of each part at most 3, satisfying max(p) <= 3 * min(p).

1, 2, 3, 4, 5, 7, 8, 10, 11, 14, 15, 19, 19, 24, 27, 31, 34, 41, 45, 53, 60, 68, 77, 89, 99, 110, 125, 138, 155, 175, 191, 212, 236, 260, 288, 320, 350, 387, 425, 468, 512, 567, 617, 679, 745, 814, 887, 974, 1058, 1160, 1263, 1376, 1494, 1632, 1769, 1922, 2086, 2262

Offset

1,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Formula

G.f.: Sum_{j>=1} q^j*(1-q^(3*j))/(1-q^j) * Product_{k=j+1..3*j} (1-q^(4*k))/(1-q^k).

Example

a(8) = 10 counts these partitions: 8, 62, 53, 44, 422, 332, 3311, 3221, 32111, 22211.

Mathematica

Nmax=60; a=Rest@CoefficientList[Series[Sum[q^j*(1-q^(3*j))/(1-q^j)*Product[(1-q^(4*k))/(1-q^k), {k, j+1, 3*j}], {j, 1, Nmax}], {q, 0, Nmax}], q] (* Vincenzo Librandi, Mar 10 2026 *)

Prog

(Magma) N := 60; R<q> := PowerSeriesRing(Integers(), N+1); S := &+[ q^j*(1-q^(3*j))/(1-q^j) * &*[(1-q^(4*k))/(1-q^k) : k in [j+1..3*j]] : j in [1..N] ]; [Coefficient(S, n) : n in [1..N]]; // Vincenzo Librandi, Mar 10 2026

Crossrefs

Cf. A393971, A393973, A394023.

Sequence in context: A107912 A190850 A066077 * A380806 A219040 A305414

Adjacent sequences: A394021 A394022 A394023 * A394025 A394026 A394027

Keyword

nonn

Author

Seiichi Manyama, Mar 07 2026

Status

approved