A329155 Expansion of Product_{k>=1} 1 / (1 - 2*x^k - 3*x^(2*k))^(1/2).

1, 1, 4, 9, 27, 67, 193, 515, 1462, 4070, 11588, 32898, 94389, 271017, 782401, 2263002, 6565987, 19086043, 55597255, 162207806, 473992799, 1386875848, 4062919108, 11915397853, 34979609583, 102781548770, 302259362326, 889566748760, 2619915414564, 7721166976185

Offset: 0

Author

Ilya Gutkovskiy, Nov 06 2019

Keywords

• nonn

Crossrefs

Cf. A002426, A067855, A081362, A242587.

Programs

Mathematica

nmax = 29; CoefficientList[Series[Product[1/(1 - 2 x^k - 3 x^(2 k))^(1/2), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 29; CoefficientList[Series[Exp[Sum[Sum[(3^d + (-1)^d)/d, {d, Divisors[k]}] x^k/2, {k, 1, nmax}]], {x, 0, nmax}], x]

Formula

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

G.f.: exp(Sum_{k>=1} ( Sum_{d|k} (3^d + (-1)^d) / d ) * x^k / 2).

G.f.: A(x) = Product_{k>=1} B(x^k), where B(x) = g.f. of A002426 (central trinomial coefficients).

a(n) ~ c * 3^(n + 1/2) / (2*sqrt(Pi*n)), where c = sqrt(Product_{k>=2} 1/((1 - 1/3^(k-1))*(1 + 1/3^k))) = sqrt(8 / (3 * QPochhammer[-1, 1/3] * QPochhammer[1/3])) = 1.23332761652608605487734981242239445... - Vaclav Kotesovec, Nov 07 2019