A394019 Expansion of g.f.: Product_{k>=1} 1 / (1 - x^(k^2 + 4)).

1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 2, 0, 1, 1, 0, 2, 0, 2, 2, 0, 2, 1, 2, 3, 0, 3, 3, 2, 3, 1, 4, 4, 2, 4, 3, 4, 5, 5, 5, 5, 4, 6, 7, 6, 6, 7, 8, 9, 6, 8, 12, 9, 11, 9, 10, 15, 10, 15, 14, 12, 17, 13, 18, 20, 13, 23, 20, 20, 23, 17, 29, 26, 22, 29, 24, 33, 32

Offset

0,14

Comments

a(n) is the number of partitions of n into parts of the form k^2+4 (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^(4/3) * sqrt(3) * Pi^(1/6) * sinh(2*Pi) * n^(7/6)) * (1 - (34 + 36*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 + 4), {k, 1, Floor[Sqrt[nmax]+1]}], {x, 0, nmax}], x]

Crossrefs

Cf. A001156, A393990, A394017, A394018, A394020.

Sequence in context: A281272 A147310 A025886 * A117355 A319571 A086966

Adjacent sequences: A394016 A394017 A394018 * A394020 A394021 A394022

Keyword

nonn

Author

Vaclav Kotesovec, Mar 06 2026

Status

approved