A394020 Expansion of g.f.: Product_{k>=1} 1 / (1 - x^(k^2 + 5)).

1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 2, 0, 1, 2, 0, 1, 2, 0, 1, 3, 1, 1, 4, 0, 2, 3, 1, 2, 5, 1, 2, 5, 1, 4, 7, 1, 4, 6, 2, 4, 9, 2, 6, 9, 2, 6, 11, 4, 8, 11, 4, 8, 15, 4, 11, 15, 6, 11, 17, 6, 14, 21, 8, 15, 23, 8, 18, 24, 11, 20, 29, 11, 22, 31

Offset

0,19

Comments

In general, if b > 0 and g.f. is Product_{k>=1} 1/(1 - x^(k^2 + b)), then a(n) ~ sqrt(b) * zeta(3/2)^(2/3) * exp(3*Pi^(1/3) * zeta(3/2)^(2/3) * n^(1/3) / 2^(4/3)) / (2^(7/3) * sqrt(3) * Pi^(1/6) * sinh(Pi*sqrt(b)) * n^(7/6)) * (1 - (34 + 9*b*Pi * zeta(1/2) * zeta(3/2)) / (9*2^(5/3) * Pi^(1/3) * zeta(3/2)^(2/3) * n^(1/3))).

a(n) is the number of partitions of n into parts of the form k^2+5 (for k>=1).

Links

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

Formula

a(n) ~ sqrt(5) * zeta(3/2)^(2/3) * exp(3*Pi^(1/3) * zeta(3/2)^(2/3) * n^(1/3) / 2^(4/3)) / (2^(7/3) * sqrt(3) * Pi^(1/6) * sinh(Pi*sqrt(5)) * n^(7/6)) * (1 - (34 + 45*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 + 5), {k, 1, Floor[Sqrt[nmax]+1]}], {x, 0, nmax}], x]

Crossrefs

Cf. A001156, A393990, A394017, A394018, A394019.

Sequence in context: A073189 A194519 A025855 * A215590 A097203 A025850

Adjacent sequences: A394017 A394018 A394019 * A394021 A394022 A394023

Keyword

nonn

Author

Vaclav Kotesovec, Mar 06 2026

Status

approved