A394018 Expansion of g.f.: Product_{k>=1} 1 / (1 - x^(k^2 + 3)).

1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 2, 0, 1, 1, 2, 0, 1, 3, 2, 1, 1, 3, 3, 1, 3, 3, 5, 1, 3, 5, 5, 3, 3, 7, 6, 3, 6, 8, 9, 3, 8, 10, 9, 6, 9, 14, 10, 8, 12, 15, 15, 9, 16, 17, 18, 13, 18, 23, 20, 17, 21, 27, 26, 19, 28, 32, 31, 23, 33, 40, 34, 30, 39, 47, 41, 36

Offset

0,13

Comments

a(n) is the number of partitions of n into parts of the form k^2+3 (for k>=1).

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Formula

a(n) ~ zeta(3/2)^(2/3) * exp(3*Pi^(1/3) * zeta(3/2)^(2/3) * n^(1/3) / 2^(4/3)) / (2^(7/3) * Pi^(1/6) * sinh(Pi*sqrt(3)) * n^(7/6)) * (1 - (34 + 27*Pi * zeta(1/2) * zeta(3/2)) / (9*2^(5/3) * Pi^(1/3) * zeta(3/2)^(2/3) * n^(1/3))).

Mathematica

nmax = 150; CoefficientList[Series[1/Product[1 - x^(k^2 + 3), {k, 1, Floor[Sqrt[nmax]+1]}], {x, 0, nmax}], x]

Crossrefs

Cf. A001156, A393990, A394017, A394019, A394020.

Sequence in context: A051010 A328342 A214269 * A130027 A116949 A359244

Adjacent sequences: A394015 A394016 A394017 * A394019 A394020 A394021

Keyword

nonn

Author

Vaclav Kotesovec, Mar 06 2026

Status

approved