A193427 G.f.: Product_{k>=1} 1/(1-x^k)^(8*k).

1, 8, 52, 272, 1266, 5344, 20992, 77584, 272727, 917936, 2975492, 9328736, 28391410, 84122688, 243265848, 688008048, 1906476351, 5184024112, 13851270944, 36409640400, 94255399886, 240529147072, 605574003464, 1505340071744

Offset

0,2

Comments

Previous name was: Number of plane partitions of n into parts of 8 kinds.

In general, if g.f. = Product_{k>=1} 1/(1-x^k)^(m*k) and m > 0, then a(n) ~ 2^(m/36 - 1/3) * exp(m/12 + 3 * 2^(-2/3) * m^(1/3) * zeta(3)^(1/3) * n^(2/3)) * (m*zeta(3))^(m/36 + 1/6) / (A^m * sqrt(3*Pi) * n^(m/36 + 2/3)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Mar 01 2015

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 19.

Formula

G.f.: Product_{k>=1} (1-x^k)^(-8*k).

a(n) ~ 2^(19/18) * zeta(3)^(7/18) * exp(2/3 + 3 * 2^(1/3) * zeta(3)^(1/3) * n^(2/3)) / (A^8 * sqrt(3*Pi) * n^(8/9)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant and zeta(3) = A002117 = 1.202056903... . - Vaclav Kotesovec, Feb 28 2015

G.f.: exp(8*Sum_{k>=1} x^k/(k*(1 - x^k)^2)). - Ilya Gutkovskiy, May 29 2018

Euler transform of 8*k. - Georg Fischer, Aug 15 2020

Maple

a:= proc(n) option remember; `if`(n=0, 1, 8*add(
      a(n-j)*numtheory[sigma][2](j), j=1..n)/n)
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Mar 11 2015

Mathematica

ANS = Block[{kmax = 50},
  Coefficient[
   Series[Product[1/(1 - x^k)^(8 k), {k, 1, kmax}], {x, 0, kmax}], x,
   Range[0, kmax]]]
(* Second program: *)
a[n_] := a[n] = If[n==0, 1, 8*Sum[a[n-j]*DivisorSigma[2, j], {j, 1, n}]/n]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Feb 07 2017, after Alois P. Heinz *)

Prog

(PARI) Vec(prod(k=1, 100\2, (1-x^k)^(-8*k), 1+O(x^101))) \\ Charles R Greathouse IV, Aug 09 2011

Crossrefs

Cf. A000219 (m=1), A161870 (m=2), A255610 (m=3), A255611 (m=4), A255612 (m=5), A255613 (m=6), A255614 (m=7).

Cf. A023007, A023003, A000712.

Column k=8 of A255961.

Sequence in context: A027225 A073377 A055283 * A022732 A256047 A227732

Adjacent sequences: A193424 A193425 A193426 * A193428 A193429 A193430

Keyword

nonn

Author

Martin Y. Veillette, Jul 28 2011

Extensions

New name from Vaclav Kotesovec, Mar 12 2015

Status

approved