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

Author

Seiichi Manyama, Nov 07 2016

Keywords

• nonn

Examples

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

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..10000
• Eric Weisstein's World of Mathematics, q-Pochhammer Symbol

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.

Programs

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 *)

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

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