A278769 Expansion of Product_{k>=1} 1/(1 - x^k)^(k*(5*k-3)/2).

1, 1, 8, 26, 88, 269, 843, 2456, 7115, 19892, 54756, 147355, 390517, 1017091, 2612670, 6617641, 16556913, 40933339, 100104289, 242276236, 580718077, 1379161494, 3247074738, 7581837910, 17564867853, 40388447308, 92206496318, 209069338580, 470944571003, 1054178579266, 2345477963043, 5188246121144, 11412352653001

Offset

0,3

Comments

Euler transform of the heptagonal numbers (A000566).

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, Heptagonal Number

Index to sequences related to polygonal numbers

Formula

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

a(n) ~ exp(-3*Zeta'(-1)/2 - 5*Zeta(3)/(8*Pi^2) - 81*Zeta(3)^3/(2*Pi^8) - 3^(13/4)*Zeta(3)^2/(2^(7/4)*Pi^5) * n^(1/4) - 3^(3/2)*Zeta(3)/(sqrt(2)*Pi^2) * sqrt(n) + 2^(7/4)*Pi/3^(5/4) * n^(3/4)) / (2^(51/32) * 3^(3/32) * Pi^(1/8) * n^(19/32)). - Vaclav Kotesovec, Dec 02 2016

Maple

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(
      d^2*(5*d-3)/2, 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 (5 k - 3)/2), {k, 1, nmax}], {x, 0, nmax}], x]

Crossrefs

Cf. A000294, A000566, A000335, A023871.

Sequence in context: A027004 A194021 A245126 * A357593 A173365 A261971

Adjacent sequences: A278766 A278767 A278768 * A278770 A278771 A278772

Keyword

nonn

Author

Ilya Gutkovskiy, Nov 28 2016

Status

approved