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 A306484 Expansion of Product_{k>=1} 1/(1 - Lucas(k)*x^k), where Lucas = A000204. 1
 1, 1, 4, 8, 24, 47, 129, 255, 641, 1308, 3064, 6225, 14286, 28792, 63571, 129240, 278329, 561044, 1190501, 2387695, 4987250, 9976529, 20536591, 40879937, 83416195, 165182927, 333581057, 658385847, 1318764282, 2590568669, 5154370637, 10082762399, 19929958391, 38848175389, 76331335061, 148233818041 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 LINKS Vaclav Kotesovec, Table of n, a(n) for n = 0..4000 FORMULA G.f.: exp(Sum_{k>=1} Sum_{j>=1} Lucas(j)^k*x^(j*k)/k). From Vaclav Kotesovec, Feb 23 2019: (Start) a(n) ~ c * 3^(n/2), where c = 27050904.849254721356174679220734831574107371522481898944915... if n is even, c = 27050894.152054775323471273913497954429537332266942696921416... if n is odd. 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. (End) MATHEMATICA nmax = 35; CoefficientList[Series[Product[1/(1 - LucasL[k] x^k), {k, 1, nmax}], {x, 0, nmax}], x] nmax = 35; CoefficientList[Series[Exp[Sum[Sum[LucasL[j]^k x^(j k)/k, {j, 1, nmax}], {k, 1, nmax}]], {x, 0, nmax}], x] 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}] CROSSREFS Cf. A000204, A261031, A300520. Sequence in context: A316961 A180002 A266821 * A208901 A319721 A115641 Adjacent sequences:  A306481 A306482 A306483 * A306485 A306486 A306487 KEYWORD nonn AUTHOR Ilya Gutkovskiy, Feb 18 2019 STATUS approved

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Last modified September 20 20:12 EDT 2019. Contains 327247 sequences. (Running on oeis4.)