A318263 Expansion of Product_{k>=1} (1 + Lucas(k)*x^k).

1, 1, 3, 7, 11, 30, 62, 129, 235, 541, 1034, 2101, 4140, 8129, 15984, 31903, 60398, 117646, 228808, 433768, 836552, 1601282, 3031299, 5736396, 10899112, 20466182, 38556342, 72522116, 135662847, 253047629, 473785878, 878655661, 1634304062, 3033385668, 5608183925

Offset

0,3

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..4500

Formula

From Vaclav Kotesovec, Aug 24 2018: (Start)

a(n) ~ c * A000032(n) * A000009(n) ~ c * phi^n * exp(Pi*sqrt(n/3)) / (4 * 3^(1/4) * n^(3/4)), where c = Product_{k>=1} ((1 + Lucas(k)/phi^k)/2) = 0.8503149035690839100210269103058319341315494385103929947491... and phi = A001622 = (1+sqrt(5))/2 is the golden ratio.

Equivalently, c = Product_{k>=1} (1 + (-1)^k/(2*phi^(2*k))),

c = 2/3 * QPochhammer[-1/2, -1/GoldenRatio^2]. (End)

Mathematica

nmax = 40; CoefficientList[Series[Product[1+LucasL[k]*x^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 40; poly = ConstantArray[0, nmax + 1]; poly[[1]] = 1; poly[[2]] = 1; Do[Do[poly[[j + 1]] += LucasL[k]*poly[[j - k + 1]], {j, nmax, k, -1}]; , {k, 2, nmax}]; poly

Crossrefs

Cf. A000032, A022629, A318248, A318264.

Sequence in context: A233517 A125879 A238673 * A213740 A267357 A152084

Adjacent sequences: A318260 A318261 A318262 * A318264 A318265 A318266

Keyword

nonn

Author

Vaclav Kotesovec, Aug 22 2018

Status

approved