OFFSET
0,1
LINKS
Eric Weisstein's World of Mathematics, Lucas Polynomial
Wikipedia, Fibonacci polynomials
FORMULA
a(n) = ((n + sqrt(n^2 + 4))^n + (n - sqrt(n^2 + 4))^n)/2^n.
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 *)
PROG
(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
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Vladimir Reshetnikov, Oct 15 2018
STATUS
approved