A279279 Expansion of Product_{k>=1} (1 + x^(k*(2*k-1))).

1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 1, 2, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 1, 2, 1, 0, 1, 1, 0, 2, 2, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 1, 1, 1, 2, 1, 0, 1, 1, 1, 1, 1, 2

Offset

0,67

Comments

Number of partitions of n into distinct hexagonal numbers (A000384).

Links

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

Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms)

Eric Weisstein's World of Mathematics, Hexagonal Number

Index to sequences related to polygonal numbers

Index entries for related partition-counting sequences

Formula

G.f.: Product_{k>=1} (1 + x^(k*(2*k-1))).

a(n) ~ zeta(3/2)^(1/3) * (sqrt(2) - 1)^(1/3) * exp(3*Pi^(1/3) * zeta(3/2)^(2/3) * (sqrt(2) - 1)^(2/3) * n^(1/3) / (2^(5/3)*d^(1/3))) / (2^(4/3 + b/(2*d)) * sqrt(3) * d^(1/6) * Pi^(1/3) * n^(5/6)) * (1 - ((sqrt(2) - 1)^(4/3) * b^2 * Pi^(2/3) * zeta(1/2) * zeta(3/2)^(1/3) / (2^(23/6) * d^(5/3)) + 5*d^(1/3) / (9 * (2*Pi)^(1/3) * (sqrt(2) - 1)^(2/3) * zeta(3/2)^(2/3))) / n^(1/3)), where d = 2, b = -1. - Vaclav Kotesovec, Mar 11 2026

Example

a(67) = 2 because we have [66, 1] and [45, 15, 6, 1].

Mathematica

nmax = 120; CoefficientList[Series[Product[1 + x^(k*(2*k-1)), {k, 1, Sqrt[1+8*nmax]/4 + 1}], {x, 0, nmax}], x] (* tuned for efficiency by Vaclav Kotesovec, Mar 10 2026 *)

Crossrefs

Cf. A000384, A024940, A033461, A218380, A278949, A279280, A279281.

Sequence in context: A230629 A027359 A236109 * A035447 A387977 A227188

Adjacent sequences: A279276 A279277 A279278 * A279280 A279281 A279282

Keyword

nonn

Author

Ilya Gutkovskiy, Dec 09 2016

Status

approved