A278949 Expansion of Product_{k>=1} 1/(1 - x^(k*(2*k-1))).

1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 7, 7, 7, 8, 9, 9, 11, 11, 11, 12, 13, 13, 15, 15, 15, 16, 17, 17, 19, 20, 20, 23, 24, 24, 26, 27, 27, 30, 31, 31, 33, 34, 35, 38, 40, 40, 44, 45, 46, 49, 51, 51, 56, 57, 58, 61, 63, 64, 69, 72, 73, 78, 80, 81, 86, 89, 90, 96, 98, 99, 105, 108, 110, 116, 120, 121, 130

Offset

0,7

Comments

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

Links

Alois P. Heinz, Table of n, a(n) for n = 0..20000

M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]

M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]

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/(1 - x^(k*(2*k-1))).

a(n) ~ Gamma(1 + b/d) * zeta(3/2)^(2/3 + b/(3*d)) * d^(1/6 + b/(3*d)) * exp(3*Pi^(1/3) * zeta(3/2)^(2/3) * n^(1/3) / (2^(4/3)*d^(1/3))) / (sqrt(3)*2^(7/3 + 2*b/(3*d)) * Pi^(7/6 - b/(6*d)) * n^(7/6 + b/(3*d))) * (1 - (136*d^2 + 120*d*b + 3*b^2*(8 - 3*Pi*zeta(1/2)*zeta(3/2))) / (72 * d^(5/3) * Pi^(1/3) * (2*zeta(3/2))^(2/3) * n^(1/3))), where d = 2, b = -1. - Vaclav Kotesovec, Mar 11 2026

Example

a(7) = 2 because we have [6, 1] and [1, 1, 1, 1, 1, 1, 1].

Maple

h:= proc(n) option remember; `if`(n<1, 0, (t->
      `if`(t*(2*t-1)>n, t-1, t))(1+h(n-1)))
    end:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1)+(t-> b(n-t, min(i, h(n-t))))(i*(2*i-1))))
    end:
a:= n-> b(n, h(n)):
seq(a(n), n=0..100);  # Alois P. Heinz, Dec 28 2018

Mathematica

nmax=90; CoefficientList[Series[Product[1/(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, A001156, A007294, A037444, A218379.

Sequence in context: A186444 A072748 A174631 * A030603 A092670 A210528

Adjacent sequences: A278946 A278947 A278948 * A278950 A278951 A278952

Keyword

nonn,easy

Author

Ilya Gutkovskiy, Dec 02 2016

Status

approved