A394017 Expansion of g.f.: Product_{k>=1} 1 / (1 - x^(k^2 + 2)).

1, 0, 0, 1, 0, 0, 2, 0, 0, 2, 0, 1, 3, 0, 1, 3, 0, 2, 5, 0, 2, 5, 1, 3, 7, 1, 3, 8, 2, 5, 10, 2, 5, 12, 3, 7, 15, 3, 9, 17, 5, 11, 20, 5, 14, 23, 7, 17, 26, 9, 20, 31, 11, 23, 36, 14, 28, 41, 17, 31, 48, 20, 38, 54, 23, 44, 63, 28, 51, 71, 31, 59, 82, 38, 68, 90

Offset

0,7

Comments

a(n) is the number of partitions of n into parts of the form k^2+2 (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^(11/6) * sqrt(3) * Pi^(1/6) * sinh(Pi*sqrt(2)) * n^(7/6)) * (1 - (34 + 18*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 + 2), {k, 1, Floor[Sqrt[nmax]+1]}], {x, 0, nmax}], x]

Crossrefs

Cf. A001156, A393990, A394018, A394019, A394020.

Sequence in context: A111775 A325166 A025844 * A035461 A118508 A029305

Adjacent sequences: A394014 A394015 A394016 * A394018 A394019 A394020

Keyword

nonn

Author

Vaclav Kotesovec, Mar 06 2026

Status

approved