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

0,1

Links

Table of n, a(n) for n=0..19.

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

Cf. A084844, A320534.

Main diagonal of A352362.

Sequence in context: A318918 A100404 A355005 * A300911 A347899 A365750

Adjacent sequences: A320567 A320568 A320569 * A320571 A320572 A320573

Keyword

nonn

Author

Vladimir Reshetnikov, Oct 15 2018

Status

approved