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A203805 G.f.: exp( Sum_{n>=1} A000204(n)^5 * x^n/n ) where A000204 is the Lucas numbers.

%I A203805 #15 Mar 27 2025 08:47:07
%S A203805 1,1,122,463,11985,85456,1262166,12018742,145326748,1540766090,
%T A203805 17495016342,191731126832,2138972609189,23652975370501,
%U A203805 262682339212290,2911255335387883,32296421465575573,358120616523262016,3971885483375619384,44047530724737577400
%N A203805 G.f.: exp( Sum_{n>=1} A000204(n)^5 * x^n/n ) where A000204 is the Lucas numbers.
%C A203805 More generally, exp(Sum_{k>=1} A000204(k)^(2*n+1) * x^k/k) = Product_{k=0..n} 1/(1 - (-1)^(n-k)*A000204(2*k+1)*x - x^2)^binomial(2*n+1,n-k).
%H A203805 G. C. Greubel, <a href="/A203805/b203805.txt">Table of n, a(n) for n = 0..950</a>
%H A203805 <a href="/index/Rec#order_32">Index entries for linear recurrences with constant coefficients</a>, signature (1,121,220,-3460,-5932,52717,52667,-483925,-81600,2532240,-1172640,-6764090,4911050,11191850,-8809960,-13039640,8809960,11191850,-4911050,-6764090,1172640,2532240,81600,-483925,-52667,52717,5932,-3460,-220,121,-1,-1).
%F A203805 G.f.: 1/( (1-x-x^2)^10 * (1+4*x-x^2)^5 * (1-11*x-x^2) ).
%F A203805 G.f.: 1/Product_{n>=1} (1 - Lucas(n)*x^n + (-1)^n*x^(2*n))^A203855(n) where A203855(n) = (1/n)*Sum_{d|n} moebius(n/d)*Lucas(d)^4.
%e A203805 G.f.: A(x) = 1 + x + 122*x^2 + 463*x^3 + 11985*x^4 + 85456*x^5 + ...
%e A203805 where
%e A203805 log(A(x)) = x + 3^5*x^2/2 + 4^5*x^3/3 + 7^5*x^4/4 + 11^5*x^5/5 + 18^5*x^6/6 + 29^5*x^7/7 + 47^5*x^8/8 + ... + Lucas(n)^5*x^n/n + ...
%t A203805 CoefficientList[Series[1/((1 - x - x^2)^10*(1 + 4*x - x^2)^5*(1 - 11*x - x^2)), {x, 0, 50}], x] (* _G. C. Greubel_, Dec 25 2017 *)
%o A203805 (PARI) /* Subroutine used in PARI programs below: */
%o A203805 {Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
%o A203805 (PARI) {a(n)=polcoeff(exp(sum(k=1, n, Lucas(k)^5*x^k/k)+x*O(x^n)), n)}
%o A203805 (PARI) {a(n,m=2)=polcoeff(prod(k=0,m, 1/(1 - (-1)^(m-k)*Lucas(2*k+1)*x - x^2+x*O(x^n))^binomial(2*m+1,m-k)),n)}
%Y A203805 Cf. A002571, A203803, A203804, A203806, A203807, A203808, A203809.
%Y A203805 Cf. A203855, A203800.
%K A203805 nonn
%O A203805 0,3
%A A203805 _Paul D. Hanna_, Jan 06 2012