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

Author

Seiichi Manyama, Nov 07 2016

Keywords

• nonn

Examples

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

Links

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

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).

Programs

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 *)

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 */

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