A162517 Triangle of coefficients of polynomials defined by Binet form: P(n,x) = ((x + d)^n - (x - d)^n)/(2*d), where d = sqrt(x+4).

0, 1, 2, 0, 3, 1, 4, 4, 4, 16, 0, 5, 10, 41, 8, 16, 6, 20, 86, 48, 96, 0, 7, 35, 161, 169, 348, 48, 64, 8, 56, 280, 456, 992, 384, 512, 0, 9, 84, 462, 1044, 2449, 1744, 2400, 256, 256, 10, 120, 732, 2136, 5482, 5920, 8640, 2560, 2560, 0, 11, 165, 1122, 4026, 11407, 16721, 26420, 14240, 14720, 1280, 1024

Offset: 1

Author

Clark Kimberling, Jul 05 2009

Keywords

• nonn
• tabf

Examples

			First six rows:
  0
  1
  2...0
  3...1...4
  4...4...16...0
  5...10..41...8...16
		

Links

• G. C. Greubel, Rows n = 1..50 of the irregular triangle, flattened

Crossrefs

Cf. A162514, A162515, A162516.

Cf. A002605, A063727.

Programs

Magma

m:=12;
Q:= func< n,x | ((x+Sqrt(x+4))^n - (x-Sqrt(x+4))^n)/(2*Sqrt(x+4)) >;
R:=PowerSeriesRing(Rationals(), m+1);
T:= func< n,k | Coefficient(R!( Q(n, x) ), n-k) >;
[0] cat [T(n,k): k in [1..n], n in [1..m]]; // G. C. Greubel, Jul 09 2023

Mathematica

Q[n_, x_]:= Q[n, x]= ((x+Sqrt[x+4])^n -(x-Sqrt[x+4])^n)/(2*Sqrt[x+4]);
T[n_, k_]:= Coefficient[Series[P[n,x], {x,0,n-k+1}], x, n-k];
Join[{0}, Table[T[n,k], {n,12}, {k,n}]//Flatten] (* G. C. Greubel, Jul 09 2023 *)

SageMath

def Q(n,x): return ((x+sqrt(x+4))^n - (x-sqrt(x+4))^n)/(2*sqrt(x+4))
def T(n,k):
    P. = PowerSeriesRing(QQ)
    return P( Q(n,x) ).list()[n-k]
[0]+flatten([[T(n,k) for k in range(1,n+1)] for n in range(1,13)]) # G. C. Greubel, Jul 09 2023

Formula

Q(n,x) = (P(n+1, x) - x*P(n,x))/(x+4), where P(n, x) is the n-th polynomial of A162516.

Q(n, x) also has the recurrence Q(n, x) = 2*x*Q(n-1, x) - (x^2 - x - 4)*Q(n-2, x).

From G. C. Greubel, Jul 09 2023: (Start)

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

Sum_{k=1..n-1} T(n, k) = A063727(n-2), n >= 2.

Sum_{k=1..n} (-1)^(k-1)*T(n, k) = A002605(n-1). (End)