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A203808 G.f.: exp( Sum_{n>=1} A000204(n)^8 * x^n/n ) where A000204 is the Lucas numbers.

%I A203808 #15 Mar 27 2025 08:56:16
%S A203808 1,1,3281,25126,6845526,121368902,12805025677,373879862237,
%T A203808 24707348223677,948781359159752,50702478932197928,2210812262034197128,
%U A203808 108528095366637700218,4974402150387759436378,236926456045384849970778,11047772769135934828000404
%N A203808 G.f.: exp( Sum_{n>=1} A000204(n)^8 * x^n/n ) where A000204 is the Lucas numbers.
%C A203808 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 A203808 G. C. Greubel, <a href="/A203808/b203808.txt">Table of n, a(n) for n = 0..596</a>
%H A203808 <a href="/index/Rec#order_256">Index entries for linear recurrences with constant coefficients</a>, order 256.
%F A203808 G.f.: 1/( (1-x)^70 * (1+3*x+x^2)^56 * (1-7*x+x^2)^28 * (1+18*x+x^2)^8 * (1-47*x+x^2) ).
%F A203808 G.f.: 1/Product_{n>=1} (1 - Lucas(n)*x^n + (-1)^n*x^(2*n))^A203858(n) where A203858(n) = (1/n)*Sum_{d|n} moebius(n/d)*Lucas(d)^7.
%e A203808 G.f.: A(x) = 1 + x + 3281*x^2 + 25126*x^3 + 6845526*x^4 + 121368902*x^5 + ...
%e A203808 where
%e A203808 log(A(x)) = x + 3^8*x^2/2 + 4^8*x^3/3 + 7^8*x^4/4 + 11^8*x^5/5 + 18^8*x^6/6 + 29^8*x^7/7 + 47^8*x^8/8 + ... + Lucas(n)^8*x^n/n + ...
%t A203808 CoefficientList[Series[1/((1 - x)^70*(1 + 3*x + x^2)^56*(1 - 7*x + x^2)^28*(1 + 18*x + x^2)^8*(1 - 47*x + x^2)), {x, 0, 50}], x] (* _G. C. Greubel_, Dec 25 2017 *)
%o A203808 (PARI) /* Subroutine used in PARI programs below: */
%o A203808 {Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
%o A203808 (PARI) {a(n)=polcoeff(exp(sum(k=1, n, Lucas(k)^8*x^k/k)+x*O(x^n)), n)}
%o A203808 (PARI) {a(n,m=4)=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 A203808 Cf. A002571, A203803, A203804, A203805, A203806, A203807, A203809.
%Y A203808 Cf. A203858, A203800.
%K A203808 nonn
%O A203808 0,3
%A A203808 _Paul D. Hanna_, Jan 06 2012