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

1,5

Comments

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

Links

Table of n, a(n) for n=1..66.

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), ...
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).

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

Cf. A118980, A104509, A309220.

Sequence in context: A192852 A152976 A153861 * A117938 A256193 A101912

Adjacent sequences: A118978 A118979 A118980 * A118982 A118983 A118984

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