A393990 Expansion of g.f.: Product_{k>=1} 1 / (1 - x^(k^2 + 1)).

1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 3, 1, 3, 1, 3, 3, 3, 4, 3, 4, 6, 4, 7, 4, 7, 7, 8, 9, 8, 9, 12, 10, 14, 10, 15, 14, 17, 18, 17, 19, 22, 21, 26, 22, 28, 27, 31, 33, 32, 35, 39, 39, 46, 41, 50, 48, 55, 57, 57, 61, 67, 67, 77, 71, 83, 82, 90, 95, 95, 102, 109, 111, 123

Offset

0,11

Comments

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

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))).

Links

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

Vaclav Kotesovec, Graph - the asymptotic ratio (200000 terms)

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) * sqrt(3) * Pi^(1/6) * sinh(Pi) * n^(7/6)) * (1 - (34 + 9*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+1), {k, 1, Floor[Sqrt[nmax]+1]}], {x, 0, nmax}], x]

Crossrefs

Cf. A001156, A393991.

Sequence in context: A240833 A110919 A281273 * A109599 A333752 A066839

Adjacent sequences: A393987 A393988 A393989 * A393991 A393992 A393993

Keyword

nonn

Author

Vaclav Kotesovec, Mar 06 2026

Status

approved