A278690 Expansion of Product_{n>=1} (1 - x^(3*n))/(1 - x^n)^2 in powers of x.
1, 2, 5, 9, 18, 31, 54, 88, 144, 225, 351, 531, 800, 1179, 1728, 2492, 3573, 5058, 7119, 9918, 13743, 18882, 25810, 35028, 47313, 63513, 84883, 112833, 149373, 196803, 258309, 337590, 439650, 570357, 737496, 950270, 1220688, 1563021, 1995642, 2540466, 3225386
Offset: 0
Examples
G.f. = 1 + 2*x + 5*x^2 + 9*x^3 + 18*x^4 + 31*x^5 + 54*x^6 + ... 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
Links
- Seiichi Manyama, Table of n, a(n) for n = 0..1000
Crossrefs
Programs
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Mathematica
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 *) a[ n_] := SeriesCoefficient[ QPochhammer[ x^3] / QPochhammer[ x]^2, {x, 0, n}]; (* Michael Somos, Nov 25 2019 *) -
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 */
Formula
G.f.: Product_{n>=1} (1 - x^(3*n))/(1 - x^n)^2.
a(n) ~ sqrt(5/3)*exp(sqrt(10*n)*Pi/3)/(12*n). - Vaclav Kotesovec, Nov 26 2016
Expansion of q^(-1/24) * eta(q^3) / eta(q)^2 in powers of q. - Michael Somos, Nov 25 2019
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