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A192382 Coefficient of x in the reduction by x^2 -> x+2 of the polynomial p(n,x) defined below in Comments.

%I A192382 #21 Mar 03 2023 16:51:36
%S A192382 0,2,4,24,80,352,1344,5504,21760,87552,349184,1398784,5591040,
%T A192382 22372352,89473024,357924864,1431633920,5726666752,22906404864,
%U A192382 91626143744,366503526400,1466016202752,5864060616704,23456250855424,93824986644480
%N A192382 Coefficient of x in the reduction by x^2 -> x+2 of the polynomial p(n,x) defined below in Comments.
%C A192382 The polynomial p(n,x) is defined by ((x+d)^n - (x-d)^n)/(2*d), where d = sqrt(x+2). For an introduction to reductions of polynomials by substitutions such as x^2 -> x+2, see A192232.
%H A192382 G. C. Greubel, <a href="/A192382/b192382.txt">Table of n, a(n) for n = 1..1000</a>
%H A192382 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,8).
%F A192382 Conjectures from _Colin Barker_, May 12 2014: (Start)
%F A192382 a(n) = 2^(n-2)*(2*(-1)^n + 2^n)/3 = 2*A003683(n-1).
%F A192382 a(n) = 2*a(n-1) + 8*a(n-2).
%F A192382 G.f.: 2*x^2 / ((1+2*x)*(1-4*x)). (End).
%F A192382 a(n) = 4^n*(1 - (-1/2)^n)/3. - _Peter Luschny_, Oct 02 2019
%F A192382 E.g.f: (1/3)*(2 + exp(2*x))*(sinh(x))^2. - _G. C. Greubel_, Feb 19 2023
%e A192382 The first five polynomials p(n,x) and their reductions are as follows:
%e A192382   p(0, x) = 1 -> 1.
%e A192382   p(1, x) = 2*x -> 2*x.
%e A192382   p(2, x) = 2 + x + 3*x^2 -> 8 + 4*x.
%e A192382   p(3, x) = 8*x + 4*x^2 + 4*x^3 -> 16 + 24*x.
%e A192382   p(4, x) = 4 + 4*x + 21*x^2 + 10*x^3 + 5*x^4 -> 96 + 80*x.
%e A192382 From these, read A083086 = (1, 0, 9, 16, 96, ...) and A192382 =(0, 2, 4, 24, 80, ...).
%p A192382 seq(4^n*(1-(-1/2)^n)/3, n=0..24); # _Peter Luschny_, Oct 02 2019
%t A192382 q[x_]:= x+2; d= Sqrt[x+2];
%t A192382 p[n_, x_]:= ((x+d)^n - (x-d)^n)/(2 d); (* suggested by A162517 *)
%t A192382 Table[Expand[p[n, x]], {n, 6}]
%t A192382 reductionRules= {x^y_?EvenQ -> q[x]^(y/2), x^y_?OddQ -> x*q[x]^((y- 1)/2)};
%t A192382 t = Table[FixedPoint[Expand[#1/. reductionRules] &, p[n,x]], {n,30}];
%t A192382 Table[Coefficient[Part[t,n], x, 0], {n,30}] (* abs value of A083086 *)
%t A192382 Table[Coefficient[Part[t,n], x, 1], {n,30}] (* 2*A003683 *)
%t A192382 Table[Coefficient[Part[t,n]/2, x, 1], {n,30}] (* A003683 *)
%t A192382 LinearRecurrence[{2,8}, {0,2}, 40] (* _G. C. Greubel_, Feb 19 2023 *)
%o A192382 (Magma) [(4^(n-1) - (-2)^(n-1))/3: n in [1..40]]; // _G. C. Greubel_, Feb 19 2023
%o A192382 (SageMath) [(4^(n-1) - (-2)^(n-1))/3 for n in range(1,41)] # _G. C. Greubel_, Feb 19 2023
%Y A192382 Cf. A003683, A083086, A162517, A192232.
%K A192382 nonn
%O A192382 1,2
%A A192382 _Clark Kimberling_, Jun 30 2011