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A203803 G.f.: exp( Sum_{n>=1} A000204(n)^3 * x^n/n ) where A000204 is the Lucas numbers.

%I A203803 #13 Mar 27 2025 08:44:17
%S A203803 1,1,14,35,205,744,3414,13926,60060,252330,1072902,4537272,19234463,
%T A203803 81452015,345084970,1461714517,6192083147,26229794928,111111714300,
%U A203803 470675847900,1993816532280,8445939457380,35777578796220,151556246864400,642002579853325,2719566542567917
%N A203803 G.f.: exp( Sum_{n>=1} A000204(n)^3 * x^n/n ) where A000204 is the Lucas numbers.
%C A203803 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 A203803 G. C. Greubel, <a href="/A203803/b203803.txt">Table of n, a(n) for n = 0..1500</a>
%H A203803 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (1,13,8,-20,-8,13,-1,-1).
%F A203803 G.f.: 1/( (1+x-x^2)^3 * (1-4*x-x^2) ).
%F A203803 G.f.: 1/Product_{n>=1} (1 - Lucas(n)*x^n + (-1)^n*x^(2*n))^A203853(n) where A203853(n) = (1/n)*Sum_{d|n} moebius(n/d)*Lucas(d)^2.
%e A203803 G.f.: A(x) = 1 + x + 14*x^2 + 35*x^3 + 205*x^4 + 744*x^5 + 3414*x^6 +...
%e A203803 where
%e A203803 log(A(x)) = x + 3^3*x^2/2 + 4^3*x^3/3 + 7^3*x^4/4 + 11^3*x^5/5 + 18^3*x^6/6 + 29^3*x^7/7 + 47^3*x^8/8 +...+ Lucas(n)^3*x^n/n +...
%t A203803 CoefficientList[Series[1/((1 + x - x^2)^3*(1 - 4*x - x^2)), {x, 0, 50}], x] (* _G. C. Greubel_, Dec 24 2017 *)
%o A203803 (PARI) /* Subroutine used in PARI programs below: */
%o A203803 {Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
%o A203803 (PARI) {a(n)=polcoeff(exp(sum(k=1, n, Lucas(k)^3*x^k/k)+x*O(x^n)), n)}
%o A203803 (PARI) {a(n,m=1)=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 A203803 Cf. A002571, A203804, A203805, A203806, A203807, A203808, A203809.
%Y A203803 Cf. A203853, A203800.
%K A203803 nonn
%O A203803 0,3
%A A203803 _Paul D. Hanna_, Jan 06 2012