OFFSET
1,5
COMMENTS
The old definition was: "Companion Pell polynomials, as a triangle."
LINKS
Wikipedia, Lucas polynomials.
FORMULA
For n >= 1, T(n,k) = Sum_{i=0..floor(k/2)} n/(n-i) * binomial(n-i,i) * binomial(n-2*i,n-k) = Sum_{i=0..floor(k/2)} (n/(n-i)) * binomial(k-i,i) * binomial(n-i,n-k). - Max Alekseyev, Oct 11 2021
G.f.: (1 + x^2)/(1-x-x^2 - x*y) (columns in reverse order). - Georg Fischer, Aug 13 2019
G.f. for row n >= 1 is the reciprocal of Lucas polynomial L_n(1+x). - Max Alekseyev, Oct 11 2021
EXAMPLE
First few rows of the triangle:
1;
1, 1;
1, 2, 3;
1, 3, 6, 4;
1, 4, 10, 12, 7;
1, 5, 15, 25, 25, 11;
...
Polynomials: (1), (x + 1), (x^2 + 2x + 3), (x^3 + 3x^2 + 6x + 4), ...
MATHEMATICA
Flatten[Map[Reverse, CoefficientList[CoefficientList[Series[(1 + x^2)/(1-x-x^2 - x*y), {x, 0, 8}], x], y]]] (* Georg Fischer, Aug 13 2019 *)
PROG
(PARI) {T(n, k) = polcoeff(polcoeff((1 + x^2)/(1 - x - x^2 - x*y) + x*O(x^n), n), n-k)}; /* Michael Somos, Oct 10 2021 */
(PARI) { A118981(n, k) = if(n==0, k==0, sum(i=0, k\2, n/(n-i) * binomial(k-i, i) * binomial(n-i, n-k) )); } \\ Max Alekseyev, Oct 11 2021
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Gary W. Adamson, May 07 2006
EXTENSIONS
Edited by N. J. A. Sloane, Aug 12 2019, replacing old definition by explicit formula from R. J. Mathar, Oct 30 2011
a(22)-a(62) from Georg Fischer, Aug 13 2019
More terms from Michel Marcus, Oct 11 2021
STATUS
approved