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A162516 Triangle of coefficients of polynomials defined by Binet form: P(n,x) = ((x+d)^n + (x-d)^n)/2, where d=sqrt(x+4).

%I A162516 #12 Jul 10 2023 01:39:32
%S A162516 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 A162516 64,1,21,119,287,644,336,448,0,1,28,182,588,1457,1360,1888,256,256,1,
%U A162516 36,270,1092,3033,4176,6240,2304,2304,0,1,45,390,1890,5925,10801,17780,11680,12160,1280,1024
%N A162516 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 A162516 G. C. Greubel, <a href="/A162516/b162516.txt">Rows n = 0..100 of triangle, flattened</a>
%F A162516 P(n,x) = 2*x*P(n-1,x) - (x^2 -x -4)*P(n-2,x).
%F A162516 From _G. C. Greubel_, Jul 09 2023: (Start)
%F A162516 T(n, k) = [x^(n-k)] ( ((x+sqrt(x+4))^n + (x-sqrt(x+4))^n)/2 ).
%F A162516 T(n, 1) = A000217(n-1), n >= 1.
%F A162516 T(n, n) = A199572(n).
%F A162516 Sum_{k=0..n} T(n, k) = A084057(n).
%F A162516 Sum_{k=0..n} 2^k*T(n, k) = A125818(n).
%F A162516 Sum_{k=0..n} (-1)^k*T(n, k) = A026150(n).
%F A162516 Sum_{k=0..n} (-2)^k*T(n, k) = A133343(n). (End)
%e A162516 First six rows:
%e A162516   1;
%e A162516   1,  0;
%e A162516   1,  1,  4;
%e A162516   1,  3, 12,  0;
%e A162516   1,  6, 25,  8, 16;
%e A162516   1, 10, 48, 40, 80, 0;
%t A162516 P[n_, x_]:= P[n, x]= ((x+Sqrt[x+4])^n + (x-Sqrt[x+4])^n)/2;
%t A162516 T[n_, k_]:= Coefficient[Series[P[n, x], {x,0,n-k+1}], x, n-k];
%t A162516 Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Jan 08 2020; Jul 09 2023 *)
%o A162516 (Magma)
%o A162516 m:=12;
%o A162516 p:= func< n,x | ((x+Sqrt(x+4))^n + (x-Sqrt(x+4))^n)/2 >;
%o A162516 R<x>:=PowerSeriesRing(Rationals(), m+1);
%o A162516 T:= func< n,k | Coefficient(R!( p(n,x) ), n-k) >;
%o A162516 [T(n,k): k in [0..n], n in [0..m]]; // _G. C. Greubel_, Jul 09 2023
%o A162516 (SageMath)
%o A162516 def p(n,x): return ((x+sqrt(x+4))^n + (x-sqrt(x+4))^n)/2
%o A162516 def T(n,k):
%o A162516     P.<x> = PowerSeriesRing(QQ)
%o A162516     return P( p(n,x) ).list()[n-k]
%o A162516 flatten([[T(n,k) for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, Jul 09 2023
%Y A162516 Cf. A162514, A162515, A162517.
%Y A162516 Cf. A000217, A026150, A084057, A125818, A199572.
%Y A162516 For fixed k, the sequences P(n,k), for n=1,2,3,4,5, are A084057, A084059, A146963, A081342, A081343, respectively.
%K A162516 nonn,tabl
%O A162516 0,6
%A A162516 _Clark Kimberling_, Jul 05 2009