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G.f.: exp( Sum_{n>=1} Lucas(n^2)*x^n/n ) where Lucas(n) = A000204(n).
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%I #11 Dec 19 2018 03:31:03

%S 1,1,4,29,585,34212,5600397,2490542953,2968152042068,9416588994339205,

%T 79216509536543420965,1762508872870620792746360,

%U 103525263562786817866762466405,16031370626878431551103688398524485

%N G.f.: exp( Sum_{n>=1} Lucas(n^2)*x^n/n ) where Lucas(n) = A000204(n).

%C Conjectured to consist entirely of integers.

%C The Lucas numbers (A000204) forms the logarithmic derivative of the Fibonacci numbers (A000045).

%C Note that Lucas(n^2) = [(1+sqrt(5))/2]^(n^2) + [(1-sqrt(5))/2]^(n^2).

%H G. C. Greubel, <a href="/A166168/b166168.txt">Table of n, a(n) for n = 0..60</a>

%H Sawian Jaidee, Patrick Moss, Tom Ward, <a href="https://arxiv.org/abs/1809.09199">Time-changes preserving zeta functions</a>, arXiv:1809.09199 [math.DS], 2018.

%F a(n) = (1/n)*Sum_{k=1..n} Lucas(k^2)*a(n-k), a(0)=1.

%F Logarithmic derivative yields A166169.

%e G.f.: A(x) = 1 + x + 4*x^2 + 29*x^3 + 585*x^4 + 34212*x^5 +...

%e log(A(x)) = x + 7*x^2/2 + 76*x^3/3 + 2207*x^4/4 + 167761*x^5/5 + 33385282*x^6/6 +...+ Lucas(n^2)*x^n/n +...

%p with(combinat): seq(coeff(series(exp(add((fibonacci(k^2-1)+fibonacci(k^2+1))*x^k/k,k=1..n)),x,n+1), x, n), n = 0 .. 15); # _Muniru A Asiru_, Dec 18 2018

%t CoefficientList[Series[Exp[Sum[LucasL[n^2]*x^n/n, {n, 1, 200}]], {x, 0, 50}], x](* _G. C. Greubel_, May 06 2016 *)

%o (PARI) {a(n)=polcoeff(exp(sum(m=1,n,(fibonacci(m^2-1)+fibonacci(m^2+1))*x^m/m)+x*O(x^n)),n)}

%Y Cf. A166169, A156216, A155200, A000204, A000045.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Oct 08 2009