A278767 Expansion of Product_{k>=1} 1/(1 - x^k)^(k*(2*k-1)).

1, 1, 7, 22, 71, 206, 616, 1712, 4743, 12677, 33407, 86085, 218677, 546060, 1345840, 3271893, 7861239, 18670881, 43883904, 102112483, 235401947, 537869136, 1218743007, 2739566083, 6111766043, 13536683750, 29775945929, 65065819486, 141285315728, 304935221675, 654318376244, 1396166024244, 2963068779402

Offset

0,3

Comments

Euler transform of the hexagonal numbers (A000384).

Links

Table of n, a(n) for n=0..32.

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]

N. J. A. Sloane, Transforms

Eric Weisstein's World of Mathematics, Hexagonal Number

Index to sequences related to polygonal numbers

Formula

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

a(n) ~ exp(-Zeta'(-1) - Zeta(3)/(2*Pi^2) - 75*Zeta(3)^3/(4*Pi^8) - 15^(5/4)*Zeta(3)^2/(2^(9/4)*Pi^5) * n^(1/4) - sqrt(15/2)*Zeta(3)/Pi^2 * sqrt(n) + 2^(9/4)*Pi/(3^(5/4)*5^(1/4)) * n^(3/4)) / (2^(67/48) * 15^(5/48) * Pi^(1/12) * n^(29/48)). - Vaclav Kotesovec, Dec 02 2016

Maple

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(
      d^2*(2*d-1), d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..35);  # Alois P. Heinz, Dec 02 2016

Mathematica

nmax=32; CoefficientList[Series[Product[1/(1 - x^k)^(k (2 k - 1)), {k, 1, nmax}], {x, 0, nmax}], x]

Crossrefs

Cf. A000294, A000384, A000335, A023871.

Sequence in context: A122238 A101289 A085287 * A286186 A282035 A302273

Adjacent sequences: A278764 A278765 A278766 * A278768 A278769 A278770

Keyword

nonn

Author

Ilya Gutkovskiy, Nov 28 2016

Status

approved