A294623 Number of partitions of n into distinct generalized heptagonal numbers (A085787).

1, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 2, 0, 2, 3, 1, 0, 3, 3, 1, 2, 2, 1, 1, 3, 3, 3, 2, 1, 2, 3, 4, 3, 2, 2, 3, 3, 3, 5, 3, 1, 3, 4, 3, 4, 5, 2, 3, 5, 4, 3, 4, 5, 4, 4, 3, 5, 5, 3, 5, 7, 5, 3, 6, 6, 6, 6, 5, 5, 6, 6, 5, 8, 7, 5, 5, 6, 7, 8, 8

Offset

0,19

Links

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

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))*(1 + x^(k*(5*k+3)/2)).

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

Example

a(18) = 2 because we have [18] and [13, 4, 1].

Mathematica

nmax = 100; CoefficientList[Series[Product[(1 + x^(k (5 k - 3)/2)) (1 + x^(k (5 k + 3)/2)), {k, 1, Floor[Sqrt[9 + 40*nmax]/10] + 1}], {x, 0, nmax}], x] (* tuned for efficiency by Vaclav Kotesovec, Mar 11 2026 *)

Crossrefs

Cf. A024940, A085787, A279280, A290942, A294621, A294624.

Sequence in context: A284582 A120336 A284558 * A337473 A039738 A191515

Adjacent sequences: A294620 A294621 A294622 * A294624 A294625 A294626

Keyword

nonn

Author

Ilya Gutkovskiy, Nov 05 2017

Status

approved