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

2, 1, 6, 36, 322, 3775, 54758, 946043, 18957314, 432083484, 11035502502, 312119004989, 9682664443202, 326872340718053, 11928306344169798, 467875943531657100, 19629328849962024962, 877095358067166709187, 41583555684469161804998, 2084882704791413248133431

Offset: 0

Author

Vladimir Reshetnikov, Oct 15 2018

Keywords

• nonn

Links

• Eric Weisstein's World of Mathematics, Lucas Polynomial
• Wikipedia, Fibonacci polynomials

Crossrefs

Cf. A084844, A320534.

Main diagonal of A352362.

Programs

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

Mathematica

Table[LucasL[n, n], {n, 0, 19}] (* or *)
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 *)

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

Formula

a(n) = ((n + sqrt(n^2 + 4))^n + (n - sqrt(n^2 + 4))^n)/2^n.