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A320570 a(n) = L_n(n), where L_n(x) is the Lucas polynomial.

%I A320570 #11 Feb 16 2025 08:33:56
%S A320570 2,1,6,36,322,3775,54758,946043,18957314,432083484,11035502502,
%T A320570 312119004989,9682664443202,326872340718053,11928306344169798,
%U A320570 467875943531657100,19629328849962024962,877095358067166709187,41583555684469161804998,2084882704791413248133431
%N A320570 a(n) = L_n(n), where L_n(x) is the Lucas polynomial.
%H A320570 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LucasPolynomial.html">Lucas Polynomial</a>
%H A320570 Wikipedia, <a href="https://en.wikipedia.org/wiki/Fibonacci_polynomials">Fibonacci polynomials</a>
%F A320570 a(n) = ((n + sqrt(n^2 + 4))^n + (n - sqrt(n^2 + 4))^n)/2^n.
%t A320570 Table[LucasL[n, n], {n, 0, 19}] (* or *)
%t A320570 Table[Round[((n + Sqrt[n^2 + 4])^n + (n - Sqrt[n^2 + 4])^n)/2^n], {n, 0, 19}] (* Round is equivalent to FullSimplify here *)
%o A320570 (PARI) for(n=0,20, print1(if(n==0,2, sum(j=0,floor(n/2), (n/(n-j))*((n-j)!*n^(n-2*j)/(j!*(n-2*j)!)))), ", ")) \\ _G. C. Greubel_, Oct 15 2018
%o A320570 (Magma) [2] cat [(&+[(n/(n-j))*(Factorial(n-j)*n^(n-2*j)/(Factorial(j)*Factorial(n-2*j))): j in [0..Floor(n/2)]]): n in [1..20]]; // _G. C. Greubel_, Oct 15 2018
%Y A320570 Cf. A084844, A320534.
%Y A320570 Main diagonal of A352362.
%K A320570 nonn
%O A320570 0,1
%A A320570 _Vladimir Reshetnikov_, Oct 15 2018