cp's OEIS Frontend

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

%I A306484 #11 Feb 23 2019 09:04:28
%S A306484 1,1,4,8,24,47,129,255,641,1308,3064,6225,14286,28792,63571,129240,
%T A306484 278329,561044,1190501,2387695,4987250,9976529,20536591,40879937,
%U A306484 83416195,165182927,333581057,658385847,1318764282,2590568669,5154370637,10082762399,19929958391,38848175389,76331335061,148233818041
%N A306484 Expansion of Product_{k>=1} 1/(1 - Lucas(k)*x^k), where Lucas = A000204.
%H A306484 Vaclav Kotesovec, <a href="/A306484/b306484.txt">Table of n, a(n) for n = 0..4000</a>
%F A306484 G.f.: exp(Sum_{k>=1} Sum_{j>=1} Lucas(j)^k*x^(j*k)/k).
%F A306484 From _Vaclav Kotesovec_, Feb 23 2019: (Start)
%F A306484 a(n) ~ c * 3^(n/2), where
%F A306484 c = 27050904.849254721356174679220734831574107371522481898944915... if n is even,
%F A306484 c = 27050894.152054775323471273913497954429537332266942696921416... if n is odd.
%F A306484 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 A306484 (End)
%t A306484 nmax = 35; CoefficientList[Series[Product[1/(1 - LucasL[k] x^k), {k, 1, nmax}], {x, 0, nmax}], x]
%t A306484 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 A306484 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 A306484 Cf. A000204, A261031, A300520.
%K A306484 nonn
%O A306484 0,3
%A A306484 _Ilya Gutkovskiy_, Feb 18 2019