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Expansion of Product_{k>=1} B(x^k), where B(x) is the g.f. of A003823.
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%I #26 May 09 2020 09:06:14

%S 1,1,1,1,2,3,3,3,4,6,7,9,10,12,14,17,21,23,26,32,40,45,51,58,69,80,89,

%T 102,116,135,154,177,198,224,253,288,326,361,408,459,521,583,650,723,

%U 812,909,1009,1122,1244,1393,1547,1716,1898,2101,2326,2575,2845,3132,3456,3809

%N Expansion of Product_{k>=1} B(x^k), where B(x) is the g.f. of A003823.

%C a(n) > 0.

%H Vaclav Kotesovec, <a href="/A327716/b327716.txt">Table of n, a(n) for n = 0..10000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Rogers-RamanujanContinuedFraction.html">Rogers-Ramanujan Continued Fraction.</a>

%F G.f.: Product_{i>=1} Product_{j>=1} (1-x^(i*(5*j-2))) * (1-x^(i*(5*j-3))) / ((1-x^(i*(5*j-1))) * (1-x^(i*(5*j-4)))).

%F G.f.: Product_{k>=1} (1-x^k)^(-A035187(k)).

%F a(n) ~ c * exp(Pi*sqrt(r*n)) / n^(3/4), where r = 4*log((1+sqrt(5))/2) / (3*sqrt(5)) = 0.2869392939760026925..., c = 0.203427046022096... - _Vaclav Kotesovec_, Sep 24 2019, updated May 09 2020

%t nmax = 60; CoefficientList[Series[Product[QPochhammer[x^(5*j - 3)] * QPochhammer[x^(5*j - 2)]/(QPochhammer[x^(5*j - 4)] * QPochhammer[x^(5*j - 1)]), {j, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Sep 23 2019 *)

%o (PARI) N=66; x='x+O('x^N); Vec(1/prod(k=1, N, (1-x^k)^sumdiv(k, d, kronecker(5, d))))

%Y Convolution inverse of A327688.

%Y Cf. A003823, A035187, A327690, A327691, A327694, A327717, A327718, A327719, A327720.

%K nonn

%O 0,5

%A _Seiichi Manyama_, Sep 23 2019