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Expansion of Product_{k>=1} ((1 + 2*x^k)/(1 - 2*x^k))^k.
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%I #14 Oct 01 2015 04:10:36

%S 1,4,16,60,192,596,1744,4892,13248,34868,89296,223660,548928,1323060,

%T 3137520,7332332,16907584,38517444,86777328,193523404,427562816,

%U 936555044,2035286576,4390850268,9409096576,20037827876,42429318480,89369282460,187325508288

%N Expansion of Product_{k>=1} ((1 + 2*x^k)/(1 - 2*x^k))^k.

%H Vaclav Kotesovec, <a href="/A261563/b261563.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) ~ c * 2^n, where c = 2 * Product_{j>=1} ((1 + 1/2^j)/(1 - 1/2^j))^(j+1) = 1021.5383556752320172813996404366861329314041364322798995039038143325883...

%F G.f.: exp( Sum_{n>=1} x^n/n * Sum_{d|n} (2^d - (-2)^d) * n^2/d^2 ). - _Paul D. Hanna_, Sep 30 2015

%t nmax = 50; CoefficientList[Series[Product[((1 + 2*x^k)/(1 - 2*x^k))^k, {k, 1, nmax}], {x, 0, nmax}], x]

%t nmax = 50; CoefficientList[Series[Exp[Sum[2^(2*k)/(2*k-1)*x^(2*k-1)/(1 - x^(2*k-1))^2, {k, 1, nmax}]], {x, 0, nmax}], x]

%o (PARI) {a(n) = polcoeff( exp( sum(m=1, n, x^m/m * sumdiv(m, d, (2^d - (-2)^d) * m^2/d^2) ) +x*O(x^n)), n)}

%o for(n=0,40,print1(a(n),", ")) \\ _Paul D. Hanna_, Sep 30 2015

%Y Cf. A156616, A261561, A261562, A261584.

%K nonn

%O 0,2

%A _Vaclav Kotesovec_, Aug 24 2015