A393147 - OEIS

A393147

Number of partitions p of n with max(p) <= 3*min(p), where the multiplicity of min(p) is 1 or 2 and all other parts are distinct.

1, 2, 2, 4, 3, 4, 5, 5, 5, 7, 8, 9, 10, 11, 13, 15, 15, 17, 20, 22, 23, 26, 29, 32, 36, 38, 43, 46, 50, 56, 62, 66, 71, 78, 84, 92, 100, 108, 117, 127, 136, 146, 159, 171, 186, 200, 214, 229, 247, 265, 285, 306, 327, 351, 378, 401, 428, 459, 490, 525, 561, 597, 636, 680, 723

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OFFSET

1,2

LINKS

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

FORMULA

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

EXAMPLE

a(8) = 5 counts these partitions: 8, 62, 53, 44, 422.

MATHEMATICA

nmax=100; gf=0; Do[p=1; Do[p=Series[p*(1+q^k), {q, 0, nmax}]//Normal; , {k, j, 3 j}]; gf=Series[gf+q^j*p, {q, 0, nmax}]//Normal; , {j, 1, nmax}]; Rest[CoefficientList[gf, q]] (* Vincenzo Librandi, Mar 14 2026 *)

PROG

(Magma) nmax := 100; R<q> := PowerSeriesRing(Integers(), nmax+5); gf := R!0; for j in [1..nmax] do p := R!1; for k in [j..3*j] do p *:= (1 + q^k); end for; gf +:= q^j * p; end for; coeff := [Coefficient(gf, n) : n in [1..nmax]]; coeff; // Vincenzo Librandi, Mar 14 2026

CROSSREFS

Cf. A384426, A385067, A393971.

Sequence in context: A109168 A015134 A171580 * A246796 A177235 A079707

Adjacent sequences: A393144 A393145 A393146 * A393148 A393149 A393150

KEYWORD

nonn

AUTHOR

Seiichi Manyama, Mar 10 2026

STATUS

approved