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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). 12

%I #12 Jul 10 2023 08:20:55

%S 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 48,64,8,56,280,456,992,384,512,0,9,84,462,1044,2449,1744,2400,256,

%U 256,10,120,732,2136,5482,5920,8640,2560,2560,0,11,165,1122,4026,11407,16721,26420,14240,14720,1280,1024

%N 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 G. C. Greubel, <a href="/A162517/b162517.txt">Rows n = 1..50 of the irregular triangle, flattened</a>

%F 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 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 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*sqrt(x+4))).

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

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

%e First six rows:

%e 0

%e 1

%e 2...0

%e 3...1...4

%e 4...4...16...0

%e 5...10..41...8...16

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

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

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

%o (Magma)

%o m:=12;

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

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

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

%o [0] cat [T(n,k): k in [1..n], n in [1..m]]; // _G. C. Greubel_, Jul 09 2023

%o (SageMath)

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

%o def T(n,k):

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

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

%o [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 Cf. A162514, A162515, A162516.

%Y Cf. A002605, A063727.

%K nonn,tabf

%O 1,3

%A _Clark Kimberling_, Jul 05 2009

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Last modified March 28 10:55 EDT 2024. Contains 371241 sequences. (Running on oeis4.)