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A306484 Expansion of Product_{k>=1} 1/(1 - Lucas(k)*x^k), where Lucas = A000204. 1

%I

%S 1,1,4,8,24,47,129,255,641,1308,3064,6225,14286,28792,63571,129240,

%T 278329,561044,1190501,2387695,4987250,9976529,20536591,40879937,

%U 83416195,165182927,333581057,658385847,1318764282,2590568669,5154370637,10082762399,19929958391,38848175389,76331335061,148233818041

%N Expansion of Product_{k>=1} 1/(1 - Lucas(k)*x^k), where Lucas = A000204.

%H Vaclav Kotesovec, <a href="/A306484/b306484.txt">Table of n, a(n) for n = 0..4000</a>

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

%F From _Vaclav Kotesovec_, Feb 23 2019: (Start)

%F a(n) ~ c * 3^(n/2), where

%F c = 27050904.849254721356174679220734831574107371522481898944915... if n is even,

%F c = 27050894.152054775323471273913497954429537332266942696921416... if n is odd.

%F In closed form, c = ((3 + sqrt(3)) * Product_{k>=3}(1/(1 - Lucas(k)/3^(k/2))) + (-1)^n * (3 - sqrt(3)) * Product_{k>=3}(1/(1 - (-1)^k*Lucas(k)/3^(k/2))))/4.

%F (End)

%t nmax = 35; CoefficientList[Series[Product[1/(1 - LucasL[k] x^k), {k, 1, nmax}], {x, 0, nmax}], x]

%t nmax = 35; CoefficientList[Series[Exp[Sum[Sum[LucasL[j]^k x^(j k)/k, {j, 1, nmax}], {k, 1, nmax}]], {x, 0, nmax}], x]

%t a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d LucasL[d]^(k/d), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 35}]

%Y Cf. A000204, A261031, A300520.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Feb 18 2019

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Last modified October 16 07:30 EDT 2019. Contains 328051 sequences. (Running on oeis4.)