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

Author

Vaclav Kotesovec, Aug 22 2018

Keywords

• nonn

Links

• 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.

Crossrefs

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

Programs

Mathematica

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

Formula

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)