%I #5 Feb 29 2024 10:47:30

%S 1,6,30,204,966,5748,29388,169944,886278,5169732,27794820,162920616,

%T 894445212,5274022920,29398573272,174041671344,980746798278,

%U 5821525480164,33071756442708,196663513473672,1124154722216244,6693497121210648,38448301937075112,229149691659210192

%N a(n) = 2^n * [x^n] Product_{k>=1} ((1 + 3*x^k)/(1 - 3*x^k))^(1/2).

%F G.f.: Product_{k>=1} ((1 + 3*(2*x)^k)/(1 - 3*(2*x)^k))^(1/2).

%F a(n) ~ c * 6^n / n^(1/2), where c = (QPochhammer(-1,1/3) / (Pi * QPochhammer(1/3)))^(1/2) = 1.333660169175690343841707335109800906849893636...

%t nmax = 30; CoefficientList[Series[Product[(1 + 3*x^k)/(1 - 3*x^k), {k, 1, nmax}]^(1/2), {x, 0, nmax}], x] * 2^Range[0, nmax]

%t nmax = 30; CoefficientList[Series[Product[(1 + 3*(2*x)^k)/(1 - 3*(2*x)^k), {k, 1, nmax}]^(1/2), {x, 0, nmax}], x]

%Y Cf. A303390, A370752.

%Y Cf. A032308, A242587.

%Y Cf. A370711, A370714.

%K nonn

%O 0,2

%A _Vaclav Kotesovec_, Feb 29 2024