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A192379 Constant term of the reduction by x^2->x+1 of the polynomial p(n,x) defined below in Comments.

%I A192379 #10 May 11 2014 11:19:50
%S A192379 1,0,5,8,45,128,505,1680,6089,21120,74909,262680,926485,3258112,
%T A192379 11474865,40382752,142171985,500432640,1761656821,6201182760,
%U A192379 21829269181,76841888640,270495370025,952182350768,3351823875225,11798909226368
%N A192379 Constant term of the reduction by x^2->x+1 of the polynomial p(n,x) defined below in Comments.
%C A192379 The polynomial p(n,x) is defined by ((x+d)^n-(x-d)^n)/(2d), where d=sqrt(x+2).  For an introduction to reductions of polynomials by substitutions such as x^2->x+1, see A192232.
%F A192379 Conjecture: a(n) = 2*a(n-1)+6*a(n-2)-2*a(n-3)-a(n-4). G.f.: -x*(x^2+2*x-1) / (x^4+2*x^3-6*x^2-2*x+1). - _Colin Barker_, May 11 2014
%e A192379 The first five polynomials p(n,x) and their reductions are as follows:
%e A192379 p(0,x)=1 -> 1
%e A192379 p(1,x)=2x -> 2x
%e A192379 p(2,x)=2+x+3x^2 -> 5+4x
%e A192379 p(3,x)=8x+4x^2+4x^3 -> 8+20x
%e A192379 p(4,x)=4+4x+21x^2+10x^3+5x^4 -> 45+60x.
%e A192379 From these, read A192379=(1,0,5,8,45,...) and A192380=(0,2,4,20,60,...).
%t A192379 q[x_] := x + 1; d = Sqrt[x + 2];
%t A192379 p[n_, x_] := ((x + d)^n - (x - d)^n )/(2 d)   (* Cf. A162517 *)
%t A192379 Table[Expand[p[n, x]], {n, 1, 6}]
%t A192379 reductionRules = {x^y_?EvenQ -> q[x]^(y/2), x^y_?OddQ -> x q[x]^((y - 1)/2)};
%t A192379 t = Table[Last[Most[FixedPointList[Expand[#1 /. reductionRules] &, p[n, x]]]], {n, 1, 30}]
%t A192379 Table[Coefficient[Part[t, n], x, 0], {n, 1, 30}]   (* A192379 *)
%t A192379 Table[Coefficient[Part[t, n], x, 1], {n, 1, 30}]   (* A192380 *)
%t A192379 Table[Coefficient[Part[t, n]/2, x, 1], {n, 1, 30}]   (* A192381 *)
%Y A192379 Cf. A192232, A192380, A192381.
%K A192379 nonn
%O A192379 1,3
%A A192379 _Clark Kimberling_, Jun 29 2011