A300520 -id:A300520 - cp's OEIS frontend

A318248 Expansion of Product_{k>=1} (1 + Fibonacci(k)*x^k).

1, 1, 1, 3, 5, 10, 18, 35, 63, 123, 220, 411, 750, 1387, 2498, 4649, 8308, 15150, 27446, 49638, 88754, 161280, 287831, 516770, 924956, 1655166, 2944850, 5272056, 9348047, 16631195, 29569572, 52421323, 92665614, 164437988, 290243745, 512649342, 904774082

Offset: 0

Vaclav Kotesovec, Aug 22 2018

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..4500
Michael Hendriksen, Nils Kapust, On the comparison of incompatibility of split systems across different taxa sizes, arXiv:2004.00062 [q-bio.PE], 2020.

Cf. A000045, A022629, A300520, A318263, A318264.

nmax = 50; CoefficientList[Series[Product[1 + Fibonacci[k]*x^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 50; poly = ConstantArray[0, nmax + 1]; poly[[1]] = 1; poly[[2]] = 1; Do[Do[poly[[j + 1]] += Fibonacci[k]*poly[[j - k + 1]], {j, nmax, k, -1}];, {k, 2, nmax}]; poly

From Vaclav Kotesovec, Aug 24 2018: (Start)
a(n) ~ c * A000045(n) * exp(r*sqrt(n)) / n^(3/4) ~ c * exp(r*sqrt(n)) * phi^n / (sqrt(5) * n^(3/4)), where r = 2*sqrt(-polylog(2, -1/sqrt(5))) = 1.273105657580344020952907652385896290122122879833..., c = 0.4521555113342405268628694407039776... and phi = A001622 = (1+sqrt(5))/2 is the golden ratio.
Equivalently, r = 2*sqrt(Pi^2/6 + log(5)^2/8 + polylog(2, -sqrt(5))). (End)

A300581 Expansion of Product_{k>=1} 1/(1 - 2^(k+1)*x^k).

Original entry on oeis.org

1, 4, 24, 112, 544, 2368, 10624, 44800, 190976, 791552, 3282944, 13414400, 54829056, 222117888, 899383296, 3625123840, 14601027584, 58659700736, 235555782656, 944552017920, 3786334535680, 15166305468416, 60736264994816, 243129089261568, 973133053952000

Offset: 0

Vaclav Kotesovec, Mar 09 2018

nonn

Cf. A075900, A300579.

Cf. A023882, A077365, A179381, A300520.

nmax = 25; CoefficientList[Series[Product[1/(1-2^(k+1)*x^k), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ c * 4^n, where c = A065446 = 1/QPochhammer(1/2) = 3.46274661945506361...

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

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Feb 18 2019

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..4000

Cf. A000204, A261031, A300520.

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}]

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)

cp's OEIS Frontend

A318248 Expansion of Product_{k>=1} (1 + Fibonacci(k)*x^k).

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A300581 Expansion of Product_{k>=1} 1/(1 - 2^(k+1)*x^k).

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

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