A277968 Expansion of ((Product_{n>=1} (1 - x^(3*n))/(1 - x^n)^3) - 1)/3 in powers of x.

0, 1, 3, 7, 16, 33, 66, 125, 231, 412, 720, 1227, 2056, 3380, 5478, 8745, 13792, 21483, 33114, 50510, 76344, 114356, 169920, 250503, 366666, 532975, 769758, 1104847, 1576640, 2237331, 3158208, 4435502, 6199479, 8624820, 11946096, 16475880, 22630864, 30962990

Offset

0,3

Links

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Formula

a(n) = A273845(n)/3, n > 0.

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

a(n) ~ exp(4*Pi*sqrt(n)/3) / (27*sqrt(2)*n^(5/4)). - Vaclav Kotesovec, Nov 10 2016

Example

G.f. = x + 3*x^2 + 7*x^3 + 16*x^4 + 33*x^5 + 66*x^6 + ...

Mathematica

nmax = 50; CoefficientList[Series[(Product[(1 - x^(3*j))/(1 - x^j)^3, {j, 1, nmax}] - 1)/3, {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 10 2016 *)
a[ n_] := SeriesCoefficient[ (QPochhammer[ x^3] / QPochhammer[ x]^3 - 1) / 3, {x, 0, n}]; (* Michael Somos, Nov 13 2016 *)

Prog

(PARI) first(n)=my(x='x); concat([0], Vec((prod(k=1, n, (1-x^(3*k))/(1-x^k)^3, 1+O(x^(n+1)))-1)/3)) \\ Charles R Greathouse IV, Nov 07 2016
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^3 + A) / eta(x + A)^3 - 1) / 3, n))}; /* Michael Somos, Nov 13 2016 */

Crossrefs

Cf. Expansion of ((Product_{n>=1} (1 - x^(k*n))/(1 - x^n)^k) - 1)/k in powers of x: A014968 (k=2), this sequence (k=3), A277974 (k=5), A160549 (k=7), A277912 (k=11).

Sequence in context: A219846 A229914 A192968 * A217942 A002936 A014668

Adjacent sequences: A277965 A277966 A277967 * A277969 A277970 A277971

Keyword

nonn

Author

Seiichi Manyama, Nov 07 2016

Status

approved