A318263 - OEIS

%I #10 Aug 24 2018 11:03:20

%S 1,1,3,7,11,30,62,129,235,541,1034,2101,4140,8129,15984,31903,60398,

%T 117646,228808,433768,836552,1601282,3031299,5736396,10899112,

%U 20466182,38556342,72522116,135662847,253047629,473785878,878655661,1634304062,3033385668,5608183925

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

%H Vaclav Kotesovec, <a href="/A318263/b318263.txt">Table of n, a(n) for n = 0..4500</a>

%F From _Vaclav Kotesovec_, Aug 24 2018: (Start)

%F 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.

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

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

%t nmax = 40; CoefficientList[Series[Product[1+LucasL[k]*x^k, {k, 1, nmax}], {x, 0, nmax}], x]

%t 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

%Y Cf. A000032, A022629, A318248, A318264.

%K nonn

%O 0,3

%A _Vaclav Kotesovec_, Aug 22 2018