A394029 Number of partitions p of n with multiplicity of each part at most 3, satisfying max(p) = 3 * min(p).

0, 0, 0, 1, 1, 2, 2, 4, 3, 5, 5, 7, 5, 6, 6, 9, 7, 10, 9, 14, 14, 17, 19, 24, 26, 30, 34, 37, 40, 48, 52, 58, 63, 68, 75, 84, 91, 101, 108, 120, 131, 145, 154, 170, 188, 208, 222, 248, 264, 294, 318, 348, 378, 417, 450, 490, 529, 576, 622, 680, 732, 795, 858, 926, 997, 1080, 1160

Offset

1,6

Links

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

Formula

G.f.: Sum_{j>=1} q^(4*j)*(1-q^(3*j))*(1-q^(9*j))/((1-q^j)*(1-q^(3*j))) * Product_{k=j+1..3*j-1} (1-q^(4*k))/(1-q^k).

Mathematica

Nmax=80; Rest@CoefficientList[Series[Sum[q^(4*j)*(1-q^(3*j))*(1-q^(9*j))/((1-q^j)*(1-q^(3*j)))*Product[(1-q^(4*k))/(1-q^k), {k, j+1, 3*j-1}], {j, 1, Nmax}], {q, 0, Nmax}]//Normal, q] (* Vincenzo Librandi, Mar 08 2026 *)

Prog

(Magma) N := 80; R<q> := PowerSeriesRing(Integers(), N+5); gf := &+[ q^(4*j)*(1-q^(3*j))*(1-q^(9*j))/((1-q^j)*(1-q^(3*j))) * &*[(1-q^(4*k))/(1-q^k) : k in [j+1..3*j-1]] : j in [1..N] ]; [Coefficient(gf, n) : n in [1..N]]; // Vincenzo Librandi, Mar 08 2026

Crossrefs

Cf. A241063, A394024, A394028.

Sequence in context: A007728 A262991 A077026 * A060026 A329437 A318632

Adjacent sequences: A394026 A394027 A394028 * A394030 A394031 A394033

Keyword

nonn

Author

Seiichi Manyama, Mar 07 2026

Status

approved