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A278690
Expansion of Product_{n>=1} (1 - x^(3*n))/(1 - x^n)^2 in powers of x.
4
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
0,2
LINKS
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
EXAMPLE
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
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 *)
PROG
(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 */
CROSSREFS
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).
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).
Cf. A298311.
Sequence in context: A069957 A091356 A107705 * A335838 A335837 A288578
KEYWORD
nonn,easy
AUTHOR
Seiichi Manyama, Nov 26 2016
STATUS
approved