A118981 - OEIS

%I #32 Feb 14 2022 00:18:37

%S 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,

%T 7,28,70,119,133,91,29,1,8,36,104,210,296,284,168,47,1,9,45,147,342,

%U 576,699,585,306,76,1,10,55,200,525,1022,1485,1580,1175,550,123

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

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

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Lucas_polynomials">Lucas polynomials</a>.

%F 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

%F G.f.: (1 + x^2)/(1-x-x^2 - x*y) (columns in reverse order). - _Georg Fischer_, Aug 13 2019

%F G.f. for row n >= 1 is the reciprocal of Lucas polynomial L_n(1+x). - _Max Alekseyev_, Oct 11 2021

%e First few rows of the triangle:

%e   1;

%e   1, 1;

%e   1, 2,  3;

%e   1, 3,  6,  4;

%e   1, 4, 10, 12,  7;

%e   1, 5, 15, 25, 25, 11;

%e   ...

%e Polynomials: (1), (x + 1), (x^2 + 2x + 3), (x^3 + 3x^2 + 6x + 4), ...

%e 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).

%t 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 *)

%o (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 */

%o (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

%Y Cf. A118980, A104509, A309220.

%K nonn,tabl

%O 1,5

%A _Gary W. Adamson_, May 07 2006

%E Edited by _N. J. A. Sloane_, Aug 12 2019, replacing old definition by explicit formula from _R. J. Mathar_, Oct 30 2011

%E a(22)-a(62) from _Georg Fischer_, Aug 13 2019

%E More terms from _Michel Marcus_, Oct 11 2021