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%I #7 Nov 07 2019 03:02:47
%S 1,1,4,9,27,67,193,515,1462,4070,11588,32898,94389,271017,782401,
%T 2263002,6565987,19086043,55597255,162207806,473992799,1386875848,
%U 4062919108,11915397853,34979609583,102781548770,302259362326,889566748760,2619915414564,7721166976185
%N Expansion of Product_{k>=1} 1 / (1 - 2*x^k - 3*x^(2*k))^(1/2).
%F G.f.: Product_{k>=1} ((1 - x^(2*k - 1)) / (1 - 3*x^k))^(1/2).
%F G.f.: exp(Sum_{k>=1} ( Sum_{d|k} (3^d + (-1)^d) / d ) * x^k / 2).
%F G.f.: A(x) = Product_{k>=1} B(x^k), where B(x) = g.f. of A002426 (central trinomial coefficients).
%F 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
%t nmax = 29; CoefficientList[Series[Product[1/(1 - 2 x^k - 3 x^(2 k))^(1/2), {k, 1, nmax}], {x, 0, nmax}], x]
%t 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]
%Y Cf. A002426, A067855, A081362, A242587.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Nov 06 2019
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