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A203804 G.f.: exp( Sum_{n>=1} A000204(n)^4 * x^n/n ) where A000204 is the Lucas numbers.

%I A203804 #14 Mar 27 2025 08:42:35
%S A203804 1,1,41,126,1526,7854,63629,400789,2870629,19254504,133376760,
%T A203804 909578760,6249172910,42785312510,293403088510,2010553849020,
%U A203804 13781960765020,94458627485820,647442212896270,4437595353800270,30415849505902910,208472981440853160,1428896115173689560
%N A203804 G.f.: exp( Sum_{n>=1} A000204(n)^4 * x^n/n ) where A000204 is the Lucas numbers.
%C A203804 More generally, exp(Sum_{k>=1} A000204(k)^(2*n) * x^k/k) = 1/(1 - (-1)^n*x)^binomial(2*n,n) * Product_{k=1..n} 1/(1 - (-1)^(n-k)*A000204(2*k)*x + x^2)^binomial(2*n,n-k).
%H A203804 G. C. Greubel, <a href="/A203804/b203804.txt">Table of n, a(n) for n = 0..1000</a>
%H A203804 <a href="/index/Rec#order_16">Index entries for linear recurrences with constant coefficients</a>, signature (1,40,45,-285,-272,1022,370,-1840,370,1022,-272,-285,45,40,1,-1).
%F A203804 G.f.: 1/( (1-x)^6 * (1+3*x+x^2)^4 * (1-7*x+x^2) ).
%F A203804 G.f.: 1/Product_{n>=1} (1 - Lucas(n)*x^n + (-1)^n*x^(2*n))^A203854(n) where A203854(n) = (1/n)*Sum_{d|n} moebius(n/d)*Lucas(d)^3.
%e A203804 G.f.: A(x) = 1 + x + 41*x^2 + 126*x^3 + 1526*x^4 + 7854*x^5 + 63629*x^6 +...
%e A203804 where
%e A203804 log(A(x)) = x + 3^4*x^2/2 + 4^4*x^3/3 + 7^4*x^4/4 + 11^4*x^5/5 + 18^4*x^6/6 + 29^4*x^7/7 + 47^4*x^8/8 +...+ Lucas(n)^4*x^n/n +...
%t A203804 CoefficientList[Series[1/((1 - x)^6*(1 + 3*x + x^2)^4*(1 - 7*x + x^2)), {x, 0, 50}], x] (* _G. C. Greubel_, Dec 24 2017 *)
%o A203804 (PARI) /* Subroutine used in PARI programs below: */
%o A203804 {Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
%o A203804 (PARI) {a(n)=polcoeff(exp(sum(k=1, n, Lucas(k)^4*x^k/k)+x*O(x^n)), n)}
%o A203804 (PARI) {a(n,m=2)=polcoeff(1/(1 - (-1)^m*x+x*O(x^n))^binomial(2*m,m) * prod(k=1,m,1/(1 - (-1)^(m-k)*Lucas(2*k)*x + x^2+x*O(x^n))^binomial(2*m,m-k)),n)}
%Y A203804 Cf. A002571, A203803, A203805, A203806, A203807, A203808, A203809.
%Y A203804 Cf. A203854, A203800.
%K A203804 nonn
%O A203804 0,3
%A A203804 _Paul D. Hanna_, Jan 06 2012