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A203806 G.f.: exp( Sum_{n>=1} A000204(n)^6 * x^n/n ) where A000204 is the Lucas numbers.

%I A203806 #15 Mar 27 2025 08:48:52
%S A203806 1,1,365,1730,97390,948562,26292937,370813165,7716851405,127699557640,
%T A203806 2397734250216,42004273130216,763345960355450,13608990417046650,
%U A203806 245008471017094450,4389301146029065420,78826300825689660420,1413927351334191841100,25376664633745265522450
%N A203806 G.f.: exp( Sum_{n>=1} A000204(n)^6 * x^n/n ) where A000204 is the Lucas numbers.
%C A203806 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 A203806 G. C. Greubel, <a href="/A203806/b203806.txt">Table of n, a(n) for n = 0..795</a>
%H A203806 <a href="/index/Rec#order_64">Index entries for linear recurrences with constant coefficients</a>, order 64.
%F A203806 G.f.: 1/( (1+x)^20 * (1-3*x+x^2)^15 * (1+7*x+x^2)^6 * (1-18*x+x^2) ).
%F A203806 G.f.: 1/Product_{n>=1} (1 - Lucas(n)*x^n + (-1)^n*x^(2*n))^A203856(n) where A203856(n) = (1/n)*Sum_{d|n} moebius(n/d)*Lucas(d)^5.
%e A203806 G.f.: A(x) = 1 + x + 365*x^2 + 1730*x^3 + 97390*x^4 + 948562*x^5 + ...
%e A203806 where
%e A203806 log(A(x)) = x + 3^6*x^2/2 + 4^6*x^3/3 + 7^6*x^4/4 + 11^6*x^5/5 + 18^6*x^6/6 + 29^6*x^7/7 + 47^6*x^8/8 + ... + Lucas(n)^6*x^n/n + ...
%t A203806 CoefficientList[Series[1/((1 + x)^20*(1 - 3*x + x^2)^15*(1 + 7*x + x^2)^6*(1 - 18*x + x^2)), {x, 0, 50}], x] (* _G. C. Greubel_, Dec 25 2017 *)
%o A203806 (PARI) /* Subroutine used in PARI programs below: */
%o A203806 {Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
%o A203806 (PARI) {a(n)=polcoeff(exp(sum(k=1, n, Lucas(k)^6*x^k/k)+x*O(x^n)), n)}
%o A203806 (PARI) {a(n,m=3)=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 A203806 Cf. A002571, A203803, A203804, A203805, A203807, A203808, A203809.
%Y A203806 Cf. A203856, A203800.
%K A203806 nonn
%O A203806 0,3
%A A203806 _Paul D. Hanna_, Jan 06 2012