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Expansion of Product_{k>=1} (1 + (x+x^2)^k) / (1 - (x+x^2)^k).
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%I #7 Dec 21 2015 14:01:35

%S 1,2,6,16,42,104,252,600,1402,3218,7282,16288,36048,78988,171516,

%T 369416,789762,1676818,3537622,7419544,15475756,32112968,66313088,

%U 136312608,279000612,568732738,1154881834,2336565080,4710930856,9466623964,18963077484,37871190504

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

%C Convolution of A266108 and A238441.

%H Vaclav Kotesovec, <a href="/A266124/b266124.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) ~ phi^n * exp(Pi*5^(-1/4)*sqrt(phi*n) + Pi^2/(40*phi)) / (8*n), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio.

%t nmax=40; CoefficientList[Series[Product[(1+(x+x^2)^k)/(1-(x+x^2)^k), {k, 1, nmax}], {x, 0, nmax}], x]

%t Table[Sum[Binomial[k, n-k] * Sum[PartitionsP[k-j]*PartitionsQ[j], {j, 0, k}], {k, 0, n}], {n, 0, 40}]

%Y Cf. A266108, A238441.

%K nonn

%O 0,2

%A _Vaclav Kotesovec_, Dec 21 2015