%I #12 Jul 10 2023 01:39:32

%S 1,1,0,1,1,4,1,3,12,0,1,6,25,8,16,1,10,45,40,80,0,1,15,75,121,252,48,

%T 64,1,21,119,287,644,336,448,0,1,28,182,588,1457,1360,1888,256,256,1,

%U 36,270,1092,3033,4176,6240,2304,2304,0,1,45,390,1890,5925,10801,17780,11680,12160,1280,1024

%N Triangle of coefficients of polynomials defined by Binet form: P(n,x) = ((x+d)^n + (x-d)^n)/2, where d=sqrt(x+4).

%H G. C. Greubel, <a href="/A162516/b162516.txt">Rows n = 0..100 of triangle, flattened</a>

%F P(n,x) = 2*x*P(n-1,x) - (x^2 -x -4)*P(n-2,x).

%F From _G. C. Greubel_, Jul 09 2023: (Start)

%F T(n, k) = [x^(n-k)] ( ((x+sqrt(x+4))^n + (x-sqrt(x+4))^n)/2 ).

%F T(n, 1) = A000217(n-1), n >= 1.

%F T(n, n) = A199572(n).

%F Sum_{k=0..n} T(n, k) = A084057(n).

%F Sum_{k=0..n} 2^k*T(n, k) = A125818(n).

%F Sum_{k=0..n} (-1)^k*T(n, k) = A026150(n).

%F Sum_{k=0..n} (-2)^k*T(n, k) = A133343(n). (End)

%e First six rows:

%e 1;

%e 1, 0;

%e 1, 1, 4;

%e 1, 3, 12, 0;

%e 1, 6, 25, 8, 16;

%e 1, 10, 48, 40, 80, 0;

%t P[n_, x_]:= P[n, x]= ((x+Sqrt[x+4])^n + (x-Sqrt[x+4])^n)/2;

%t T[n_, k_]:= Coefficient[Series[P[n, x], {x,0,n-k+1}], x, n-k];

%t Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Jan 08 2020; Jul 09 2023 *)

%o (Magma)

%o m:=12;

%o p:= func< n,x | ((x+Sqrt(x+4))^n + (x-Sqrt(x+4))^n)/2 >;

%o R<x>:=PowerSeriesRing(Rationals(), m+1);

%o T:= func< n,k | Coefficient(R!( p(n,x) ), n-k) >;

%o [T(n,k): k in [0..n], n in [0..m]]; // _G. C. Greubel_, Jul 09 2023

%o (SageMath)

%o def p(n,x): return ((x+sqrt(x+4))^n + (x-sqrt(x+4))^n)/2

%o def T(n,k):

%o P.<x> = PowerSeriesRing(QQ)

%o return P( p(n,x) ).list()[n-k]

%o flatten([[T(n,k) for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, Jul 09 2023

%Y Cf. A162514, A162515, A162517.

%Y Cf. A000217, A026150, A084057, A125818, A199572.

%Y For fixed k, the sequences P(n,k), for n=1,2,3,4,5, are A084057, A084059, A146963, A081342, A081343, respectively.

%K nonn,tabl

%O 0,6

%A _Clark Kimberling_, Jul 05 2009