A118981 Triangle read by rows: T(n,k) = abs( A104509(n-1,n-k) ).

1, 1, 1, 1, 2, 3, 1, 3, 6, 4, 1, 4, 10, 12, 7, 1, 5, 15, 25, 25, 11, 1, 6, 21, 44, 60, 48, 18, 1, 7, 28, 70, 119, 133, 91, 29, 1, 8, 36, 104, 210, 296, 284, 168, 47, 1, 9, 45, 147, 342, 576, 699, 585, 306, 76, 1, 10, 55, 200, 525, 1022, 1485, 1580, 1175, 550, 123

Offset: 1

Author

Gary W. Adamson, May 07 2006

Keywords

• nonn
• tabl

Comments

The old definition was: "Companion Pell polynomials, as a triangle."

Examples

			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), ...
Row 3: (1, 2, 3); as (x^2 + 2x + 3) = f(x), (x=1,2,3,...) of column 3 of A309220: (6, 11, 18, 27, 38, 51,...). The latter sequence = binomial transform of row 3 of A118980: (6, 5, 2).
		

Links

• Wikipedia, Lucas polynomials.

Crossrefs

Cf. A118980, A104509, A309220.

Programs

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 *)

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

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

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