OFFSET
0,1
LINKS
Eric Weisstein's World of Mathematics, Lucas Polynomial
FORMULA
T(n, k) = Sum_{j=0..floor(k/2)} binomial(k-j, j)*(k/(k-j))*n^(k-2*j) for k >= 1.
T(n, k) = (n/2 + sqrt((n/2)^2 + 1))^k + (n/2 - sqrt((n/2)^2 + 1))^k.
T(n, k) = [x^k] ((2 - n*x)/(1 - n*x - x^2)).
T(n, k) = n^k*hypergeom([1/2 - k/2, -k/2], [1 - k], -4/n^2) for n,k >= 1.
EXAMPLE
Array starts:
n\k 0, 1, 2, 3, 4, 5, 6, 7, 8, ...
--------------------------------------------------------------
[0] 2, 0, 2, 0, 2, 0, 2, 0, 2, ... A010673
[1] 2, 1, 3, 4, 7, 11, 18, 29, 47, ... A000032
[2] 2, 2, 6, 14, 34, 82, 198, 478, 1154, ... A002203
[3] 2, 3, 11, 36, 119, 393, 1298, 4287, 14159, ... A006497
[4] 2, 4, 18, 76, 322, 1364, 5778, 24476, 103682, ... A014448
[5] 2, 5, 27, 140, 727, 3775, 19602, 101785, 528527, ... A087130
[6] 2, 6, 38, 234, 1442, 8886, 54758, 337434, 2079362, ... A085447
[7] 2, 7, 51, 364, 2599, 18557, 132498, 946043, 6754799, ... A086902
[8] 2, 8, 66, 536, 4354, 35368, 287298, 2333752, 18957314, ... A086594
[9] 2, 9, 83, 756, 6887, 62739, 571538, 5206581, 47430767, ... A087798
MAPLE
T := (n, k) -> (n/2 + sqrt((n/2)^2 + 1))^k + (n/2 - sqrt((n/2)^2 + 1))^k:
seq(seq(simplify(T(n - k, k)), k = 0..n), n = 0..10);
MATHEMATICA
Table[LucasL[k, n], {n, 0, 9}, {k, 0, 9}] // TableForm
(* or *)
T[ 0, k_] := 2 Mod[k+1, 2]; T[n_, 0] := 2;
T[n_, k_] := n^k Hypergeometric2F1[1/2 - k/2, -k/2, 1 - k, -4/n^2];
Table[T[n, k], {n, 0, 9}, {k, 0, 8}] // TableForm
PROG
(PARI)
T(n, k) = ([0, 1; 1, k]^n*[2; k])[1, 1] ;
export(T)
for(k = 0, 9, print(parvector(10, n, T(n - 1, k))))
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Peter Luschny, Mar 18 2022
STATUS
approved