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Expansion of Product_{k>=1} 1/(1 - x^k * (k + x)).
6

%I #19 May 06 2021 06:00:14

%S 1,1,4,9,22,47,107,221,468,953,1932,3814,7560,14625,28192,53757,

%T 101827,190907,356362,659716,1215314,2224968,4053914,7346367,13260001,

%U 23822114,42629786,75991017,134991954,238948942,421656911,741750026,1301116634,2275985891,3971022904

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

%H Seiichi Manyama, <a href="/A336975/b336975.txt">Table of n, a(n) for n = 0..1000</a>

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

%F a(n) ~ c * n * phi^(n+1) / 5, where c = Product_{k>=3} 1/(1 - 1/phi^k*(k + 1/phi)) = 167.5661037860673786430316975350024960626825333609486463342... and phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - _Vaclav Kotesovec_, May 06 2021

%t m = 34; CoefficientList[Series[Product[1/(1 - x^k*(k + x)), {k, 1, m}], {x, 0, m}], x] (* _Amiram Eldar_, May 01 2021 *)

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

%o (PARI) N=66; x='x+O('x^N); Vec(exp(sum(k=1, N, x^k*sumdiv(k, d, (k/d+x)^d/d))))

%Y Cf. A006906, A227681, A336976, A336977, A336978, A336979, A336980.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Aug 09 2020