A277968 - OEIS

%I #42 Nov 13 2016 16:23:29

%S 0,1,3,7,16,33,66,125,231,412,720,1227,2056,3380,5478,8745,13792,

%T 21483,33114,50510,76344,114356,169920,250503,366666,532975,769758,

%U 1104847,1576640,2237331,3158208,4435502,6199479,8624820,11946096,16475880,22630864,30962990

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

%H Seiichi Manyama, <a href="/A277968/b277968.txt">Table of n, a(n) for n = 0..1000</a>

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

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

%F a(n) ~ exp(4*Pi*sqrt(n)/3) / (27*sqrt(2)*n^(5/4)). - _Vaclav Kotesovec_, Nov 10 2016

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

%t 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 *)

%t a[ n_] := SeriesCoefficient[ (QPochhammer[ x^3] / QPochhammer[ x]^3 - 1) / 3, {x, 0, n}]; (* _Michael Somos_, Nov 13 2016 *)

%o (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

%o (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 */

%Y 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).

%K nonn

%O 0,3

%A _Seiichi Manyama_, Nov 07 2016