A279280 Expansion of Product_{k>=1} (1 + x^(k*(5*k-3)/2)).

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

Offset

0,82

Comments

Number of partitions of n into distinct heptagonal numbers (A000566).

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..20000

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

Eric Weisstein's World of Mathematics, Heptagonal Number

Index to sequences related to polygonal numbers

Index entries for related partition-counting sequences

Formula

G.f.: Product_{k>=1} (1 + x^(k*(5*k-3)/2)).

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 = 5/2, b = -3/2. - Vaclav Kotesovec, Mar 11 2026

Example

a(81) = 2 because we have [81] and [55, 18, 7, 1].

Mathematica

nmax = 120; CoefficientList[Series[Product[1 + x^(k*(5*k-3)/2), {k, 1, Sqrt[9 + 40 nmax]/10 + 1}], {x, 0, nmax}], x] (* tuned for efficiency by Vaclav Kotesovec, Mar 10 2026 *)

Crossrefs

Cf. A000566, A024940, A033461, A218380, A279012, A279279, A279281.

Sequence in context: A263338 A103522 A101108 * A284093 A284095 A279593

Adjacent sequences: A279277 A279278 A279279 * A279281 A279282 A279283

Keyword

nonn

Author

Ilya Gutkovskiy, Dec 09 2016

Status

approved