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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).

%I A162517 #12 Jul 10 2023 08:20:55
%S A162517 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,
%T A162517 48,64,8,56,280,456,992,384,512,0,9,84,462,1044,2449,1744,2400,256,
%U A162517 256,10,120,732,2136,5482,5920,8640,2560,2560,0,11,165,1122,4026,11407,16721,26420,14240,14720,1280,1024
%N 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).
%H A162517 G. C. Greubel, <a href="/A162517/b162517.txt">Rows n = 1..50 of the irregular triangle, flattened</a>
%F A162517 Q(n,x) = (P(n+1, x) - x*P(n,x))/(x+4), where P(n, x) is the n-th polynomial of A162516.
%F A162517 Q(n, x) also has the recurrence Q(n, x) = 2*x*Q(n-1, x) - (x^2 - x - 4)*Q(n-2, x).
%F A162517 From _G. C. Greubel_, Jul 09 2023: (Start)
%F A162517 T(n, k) = [x^(n-k)](((x+sqrt(x+4))^n -(x-sqrt(x+4))^n)/(2*sqrt(x+4))).
%F A162517 Sum_{k=1..n-1} T(n, k) = A063727(n-2), n >= 2.
%F A162517 Sum_{k=1..n} (-1)^(k-1)*T(n, k) = A002605(n-1). (End)
%e A162517 First six rows:
%e A162517   0
%e A162517   1
%e A162517   2...0
%e A162517   3...1...4
%e A162517   4...4...16...0
%e A162517   5...10..41...8...16
%t A162517 Q[n_, x_]:= Q[n, x]= ((x+Sqrt[x+4])^n -(x-Sqrt[x+4])^n)/(2*Sqrt[x+4]);
%t A162517 T[n_, k_]:= Coefficient[Series[P[n,x], {x,0,n-k+1}], x, n-k];
%t A162517 Join[{0}, Table[T[n,k], {n,12}, {k,n}]//Flatten] (* _G. C. Greubel_, Jul 09 2023 *)
%o A162517 (Magma)
%o A162517 m:=12;
%o A162517 Q:= func< n,x | ((x+Sqrt(x+4))^n - (x-Sqrt(x+4))^n)/(2*Sqrt(x+4)) >;
%o A162517 R<x>:=PowerSeriesRing(Rationals(), m+1);
%o A162517 T:= func< n,k | Coefficient(R!( Q(n, x) ), n-k) >;
%o A162517 [0] cat [T(n,k): k in [1..n], n in [1..m]]; // _G. C. Greubel_, Jul 09 2023
%o A162517 (SageMath)
%o A162517 def Q(n,x): return ((x+sqrt(x+4))^n - (x-sqrt(x+4))^n)/(2*sqrt(x+4))
%o A162517 def T(n,k):
%o A162517     P.<x> = PowerSeriesRing(QQ)
%o A162517     return P( Q(n,x) ).list()[n-k]
%o A162517 [0]+flatten([[T(n,k) for k in range(1,n+1)] for n in range(1,13)]) # _G. C. Greubel_, Jul 09 2023
%Y A162517 Cf. A162514, A162515, A162516.
%Y A162517 Cf. A002605, A063727.
%K A162517 nonn,tabf
%O A162517 1,3
%A A162517 _Clark Kimberling_, Jul 05 2009