A278768 Expansion of Product_{k>=1} 1/(1 - x^k)^(k*(3*k-1)/2).

1, 1, 6, 18, 55, 150, 424, 1113, 2923, 7401, 18510, 45271, 109297, 259447, 608428, 1407958, 3222132, 7292198, 16340830, 36265672, 79775931, 173999194, 376497975, 808471181, 1723592762, 3649271887, 7675809680, 16043777217, 33332888108, 68853608216, 141438908854, 288994878713, 587458691042

Offset

0,3

Comments

Euler transform of the pentagonal numbers (A000326).

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000

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, Pentagonal Number

Index to sequences related to polygonal numbers

Formula

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

a(n) ~ exp(-Zeta'(-1)/2 - 3*Zeta(3)/(8*Pi^2) - 25*Zeta(3)^3/(6*Pi^8) - 5^(5/4)*Zeta(3)^2/(2^(7/4)*Pi^5) * n^(1/4) - sqrt(5/2)*Zeta(3)/Pi^2 * sqrt(n) + 2^(7/4)*Pi/(3*5^(1/4)) * n^(3/4)) / (2^(155/96) * 5^(11/96) * Pi^(1/24) * n^(59/96)). - Vaclav Kotesovec, Dec 02 2016

Maple

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

Crossrefs

Cf. A000294, A000326, A000335, A023871.

Sequence in context: A292295 A183913 A056349 * A035070 A075386 A056343

Adjacent sequences: A278765 A278766 A278767 * A278769 A278770 A278771

Keyword

nonn

Author

Ilya Gutkovskiy, Nov 28 2016

Status

approved