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A192376 Constant term of the reduction by x^2->x+2 of the polynomial p(n,x) defined below in Comments.

%I A192376 #10 May 11 2014 11:19:45
%S A192376 1,0,7,16,73,256,975,3616,13521,50432,188247,702512,2621849,9784832,
%T A192376 36517535,136285248,508623521,1898208768,7084211623,26438637648,
%U A192376 98670339049,368242718464,1374300534895,5128959421024,19141537149297,71437189176064
%N A192376 Constant term of the reduction by x^2->x+2 of the polynomial p(n,x) defined below in Comments.
%C A192376 The polynomial p(n,x) is defined by ((x+d)^n-(x-d)^n)/(2d), where d=sqrt(x+1).  For an introduction to reductions of polynomials by substitutions such as x^2->x+1, see A192232.
%F A192376 Conjecture: a(n) = 2*a(n-1)+6*a(n-2)+2*a(n-3)-a(n-4). G.f.: x*(x-1)^2 / ((x+1)^2*(x^2-4*x+1)). - _Colin Barker_, May 11 2014
%e A192376 The first five polynomials p(n,x) and their reductions are as follows:
%e A192376 p(0,x)=1 -> 1
%e A192376 p(1,x)=2x -> 2x
%e A192376 p(2,x)=4+x+3x^2 -> 7+4x
%e A192376 p(3,x)=16x+4x^2+4x^3 -> 16+20x
%e A192376 p(4,x)=16+8x+41x^2+10x^3+5x^4 -> 73+68x.
%e A192376 From these, read A192376=(1,0,7,16,73,...) and A192377=(0,2,4,20,68,...).
%t A192376 q[x_] := x + 2; d = Sqrt[x + 1];
%t A192376 p[n_, x_] := ((x + d)^n - (x - d)^ n )/(2 d)   (* Cf. A162517 *)
%t A192376 Table[Expand[p[n, x]], {n, 1, 6}]
%t A192376 reductionRules = {x^y_?EvenQ -> q[x]^(y/2), x^y_?OddQ -> x q[x]^((y - 1)/2)};
%t A192376 t = Table[Last[Most[FixedPointList[Expand[#1 /. reductionRules] &, p[n, x]]]], {n, 1,  30}]
%t A192376 Table[Coefficient[Part[t, n], x, 0], {n, 1, 30}] (* A192376 *)
%t A192376 Table[Coefficient[Part[t, n], x, 1], {n, 1, 30}] (* A192377 *)
%t A192376 Table[Coefficient[Part[t, n]/2, x, 1], {n, 1, 30}] (* A192378 *)
%Y A192376 Cf. A192232, A192377, A192378.
%K A192376 nonn
%O A192376 1,3
%A A192376 _Clark Kimberling_, Jun 29 2011