A273845 Expansion of Product_{n>=1} (1 - x^(3*n))/(1 - x^n)^3 in powers of x.

1, 3, 9, 21, 48, 99, 198, 375, 693, 1236, 2160, 3681, 6168, 10140, 16434, 26235, 41376, 64449, 99342, 151530, 229032, 343068, 509760, 751509, 1099998, 1598925, 2309274, 3314541, 4729920, 6711993, 9474624, 13306506, 18598437, 25874460, 35838288, 49427640, 67892592

Offset

0,2

Links

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Eric Weisstein's World of Mathematics, q-Pochhammer Symbol

Formula

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

a(n) ~ exp(4*Pi*sqrt(n)/3) / (9*sqrt(2)*n^(5/4)). - Vaclav Kotesovec, Nov 10 2016

G.f.: (x^3; x^3)_inf/((x; x)_inf)^3, where (a; q)_inf is the q-Pochhammer symbol. - Vladimir Reshetnikov, Nov 20 2016

a(0) = 1, a(n) = (3/n)*Sum_{k=1..n} A078708(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 29 2017

It appears that the g.f. A(x) = F(x)^3, where F(x) = exp( Sum_{n >= 0} x^(3*n+1)/((3*n + 1)*(1 - x^(3*n+1))) + x^(3*n+2)/((3*n + 2)*(1 - x^(3*n + 2))) ). Cf. A132972. - Peter Bala, Dec 23 2021

Example

G.f.: 1 + 3*x + 9*x^2 + 21*x^3 + 48*x^4 + 99*x^5 + 198*x^6 + ...

Mathematica

nmax = 50; CoefficientList[Series[Product[(1 - x^(3*k))/(1 - x^k)^3, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 10 2016 *)
(QPochhammer[x^3, x^3]/QPochhammer[x, x]^3 + O[x]^40)[[3]] (* Vladimir Reshetnikov, Nov 20 2016 *)

Prog

(PARI) first(n)=my(x='x); Vec(prod(k=1, n, (1-x^(3*k))/(1-x^k)^3, 1+O(x^(n+1)))) \\ Charles R Greathouse IV, Nov 07 2016
(PARI) lista(nn) = {q='q+O('q^nn); Vec(eta(q^3)/eta(q)^3)} \\ Altug Alkan, Mar 20 2018

Crossrefs

Expansion of Product_{n>=1} (1 - x^(k*n))/(1 - x^n)^k in powers of x: A015128 (k=2), this sequence (k=3), A274327 (k=4), A277212 (k=5), A277283 (k=6), A160539 (k=7).

Cf. A005928, A132972.

Sequence in context: A014286 A000714 A267226 * A090984 A319781 A006813

Adjacent sequences: A273842 A273843 A273844 * A273846 A273847 A273848

Keyword

nonn

Author

Seiichi Manyama, Nov 07 2016

Status

approved