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A192373 Constant term in the reduction of the polynomial p(n,x) defined at A162517 and below in Comments.

%I A192373 #21 May 12 2014 08:59:11
%S A192373 1,0,7,8,77,192,1043,3472,15529,57792,240655,934808,3789653,14963328,
%T A192373 60048443,238578976,953755537,3798340224,15162325975,60438310184,
%U A192373 241126038941,961476161856,3835121918243,15294304429744,61000836720313,243280700771904
%N A192373 Constant term in the reduction of the polynomial p(n,x) defined at A162517 and below in Comments.
%C A192373 The polynomials are given by p(n,x)=((x+d)^n-(x-d)^n)/(2d), where d=sqrt(x+4).
%C A192373 For an introduction to reductions of polynomials by substitutions such as x^2->x+1, see A192232.
%F A192373 Conjecture: a(n) = 2*a(n-1)+10*a(n-2)-6*a(n-3)-9*a(n-4). G.f.: -x*(x+1)*(3*x-1) / (9*x^4+6*x^3-10*x^2-2*x+1). - _Colin Barker_, May 09 2014
%e A192373 The first five polynomials p(n,x) and their reductions are as follows:
%e A192373 p(0,x)=1 -> 1
%e A192373 p(1,x)=2x -> 2x
%e A192373 p(2,x)=4+x+3x^2 -> 7+4x
%e A192373 p(3,x)=16x+4x^2+4x^3 -> 8+28x
%e A192373 p(4,x)=16+8x+41x^2+10x^3+5x^4 -> 77+84x.
%e A192373 From these, read A192352=(1,0,7,8,77,...) and A049602=(0,2,4,28,84,...).
%t A192373 q[x_] := x + 1; d = Sqrt[x + 4];
%t A192373 p[n_, x_] := ((x + d)^n - (x - d)^n )/(2 d) (* A162517 *)
%t A192373 Table[Expand[p[n, x]], {n, 1, 6}]
%t A192373 reductionRules = {x^y_?EvenQ -> q[x]^(y/2), x^y_?OddQ -> x q[x]^((y - 1)/2)}; t = Table[Last[Most[FixedPointList[Expand[#1 /. reductionRules] &, p[n, x]]]], {n, 1, 30}]
%t A192373 Table[Coefficient[Part[t, n], x, 0], {n, 1, 30}]  (* A192373 *)
%t A192373 Table[Coefficient[Part[t, n], x, 1], {n, 1, 30}]  (* A192374 *)
%t A192373 Table[Coefficient[Part[t, n]/2, x, 1], {n, 1, 30}] (* A192375 *)
%Y A192373 Cf. A192232, A192374, A192375, A162517.
%K A192373 nonn
%O A192373 1,3
%A A192373 _Clark Kimberling_, Jun 29 2011