A278690 - OEIS

%I #21 Nov 17 2020 11:04:05

%S 1,2,5,9,18,31,54,88,144,225,351,531,800,1179,1728,2492,3573,5058,

%T 7119,9918,13743,18882,25810,35028,47313,63513,84883,112833,149373,

%U 196803,258309,337590,439650,570357,737496,950270,1220688,1563021,1995642,2540466,3225386

%N Expansion of Product_{n>=1} (1 - x^(3*n))/(1 - x^n)^2 in powers of x.

%H Seiichi Manyama, <a href="/A278690/b278690.txt">Table of n, a(n) for n = 0..1000</a>

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

%F a(n) ~ sqrt(5/3)*exp(sqrt(10*n)*Pi/3)/(12*n). - _Vaclav Kotesovec_, Nov 26 2016

%F Expansion of q^(-1/24) * eta(q^3) / eta(q)^2 in powers of q. - _Michael Somos_, Nov 25 2019

%F G.f.: 1/Product_{n > = 1} ( 1 - x^(n/gcd(n,k)) ) for k = 3. Cf. A000041 (k = 1), A015128 (k = 2), A298311 (k = 4) and A160461 (k = 5). - _Peter Bala_, Nov 17 2020

%e G.f. = 1 + 2*x + 5*x^2 + 9*x^3 + 18*x^4 + 31*x^5 + 54*x^6 + ...

%e G.f. = q + 2*q^25 + 5*q^49 + 9*q^73 + 18*q^97 + 31*q^121 + 54*q^145 + ... - _Michael Somos_, Nov 25 2019

%t nmax = 50; CoefficientList[Series[Product[(1 - x^(3*k))/(1 - x^k)^2, {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Nov 26 2016 *)

%t a[ n_] := SeriesCoefficient[ QPochhammer[ x^3] / QPochhammer[ x]^2, {x, 0, n}]; (* _Michael Somos_, Nov 25 2019 *)

%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^3 + A) / eta(x + A)^2, n))}; /* _Michael Somos_, Nov 25 2019 */

%Y Cf. Product_{n>=1} (1 - x^(3*n))/(1 - x^n)^k: A000726 (k=1), this sequence (k=2), A273845 (k=3), A182819 (k=4).

%Y Cf. Product_{n>=1} (1 - x^(k*n))/(1 - x^n)^2: A000041 (k=1), A015128 (k=2), this sequence (k=3), A160461 (k=5).

%Y Cf. A298311.

%K nonn,easy

%O 0,2

%A _Seiichi Manyama_, Nov 26 2016