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A203807 G.f.: exp( Sum_{n>=1} A000204(n)^7 * x^n/n ) where A000204 is the Lucas numbers.

%I A203807 #15 Mar 27 2025 08:52:07
%S A203807 1,1,1094,6555,809765,10676072,570282082,11680775298,427757608420,
%T A203807 10880625876510,341910837405634,9500984180929624,282684350289144641,
%U A203807 8100555748749977985,236841648715969283630,6851665210550903756723,199305150210062939465293
%N A203807 G.f.: exp( Sum_{n>=1} A000204(n)^7 * x^n/n ) where A000204 is the Lucas numbers.
%C A203807 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 A203807 G. C. Greubel, <a href="/A203807/b203807.txt">Table of n, a(n) for n = 0..680</a>
%H A203807 <a href="/index/Rec#order_128">Index entries for linear recurrences with constant coefficients</a>, order 128.
%F A203807 G.f.: 1/( (1+x-x^2)^35 * (1-4*x-x^2)^21 * (1+11*x-x^2)^7 * (1-29*x-x^2) ).
%F A203807 G.f.: 1/Product_{n>=1} (1 - Lucas(n)*x^n + (-1)^n*x^(2*n))^A203857(n) where A203857(n) = (1/n)*Sum_{d|n} moebius(n/d)*Lucas(d)^6.
%e A203807 G.f.: A(x) = 1 + x + 1094*x^2 + 6555*x^3 + 809765*x^4 + 10676072*x^5 + ...
%e A203807 where
%e A203807 log(A(x)) = x + 3^7*x^2/2 + 4^7*x^3/3 + 7^7*x^4/4 + 11^7*x^5/5 + 18^7*x^6/6 + 29^7*x^7/7 + 47^7*x^8/8 + ... + Lucas(n)^7*x^n/n + ...
%t A203807 CoefficientList[Series[1/((1 + x - x^2)^35*(1 - 4*x - x^2)^21*(1 + 11*x - x^2)^7*(1 - 29*x - x^2)), {x, 0, 50}], x] (* _G. C. Greubel_, Dec 25 2017 *)
%o A203807 (PARI) /* Subroutine used in PARI programs below: */
%o A203807 {Lucas(n)=fibonacci(n-1)+fibonacci(n+1)}
%o A203807 (PARI) {a(n)=polcoeff(exp(sum(k=1, n, Lucas(k)^7*x^k/k)+x*O(x^n)), n)}
%o A203807 (PARI) {a(n,m=3)=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 A203807 Cf. A002571, A203803, A203804, A203805, A203806, A203808, A203809.
%Y A203807 Cf. A203857, A203800.
%K A203807 nonn
%O A203807 0,3
%A A203807 _Paul D. Hanna_, Jan 06 2012