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A192352 Constant term of the reduction of the polynomial p(n,x)=(1/2)((x+1)^n+(x-1)^n) by x^2->x+1.

%I A192352 #7 Sep 11 2012 17:49:43
%S A192352 1,0,2,1,9,13,51,106,322,771,2135,5401,14445,37324,98514,256621,
%T A192352 673933,1760997,4615823,12075526,31628466,82781215,216761547,
%U A192352 567428401,1485645049,3889310328,10182603746,26657986681,69792188337,182717232061
%N A192352 Constant term of the reduction of the polynomial p(n,x)=(1/2)((x+1)^n+(x-1)^n) by x^2->x+1.
%C A192352 1
%F A192352 Empirical G.f.: -x*(x^3-x^2-2*x+1)/((x^2-3*x+1)*(x^2-x-1)). [_Colin Barker_, Sep 11 2012]
%e A192352 For an introduction to reductions of polynomials by substitutions such as x^2->x+1, see A192232.
%e A192352 The first six polynomials p(n,x) and their reductions are as follows:
%e A192352 p(0,x)=1 -> 1
%e A192352 p(1,x)=x -> x
%e A192352 p(2,x)=1+x^2 -> 2+x
%e A192352 p(3,x)=3x+x^3 -> 1+5x
%e A192352 p(4,x)=1+6x^2+x^4 -> 9+9x
%e A192352 p(5,x)=5x+10x^3+x^5 -> 13+30x.
%e A192352 From these, we read
%e A192352 A192352=(1,0,2,1,9,13...) and A049602=(0,1,1,5,9,30...).
%t A192352 q[x_] := x + 1; d = 1;
%t A192352 p[n_, x_] := ((x + d)^n + (x - d)^n )/
%t A192352   2 (* similar to polynomials defined at A161516 *)
%t A192352 Table[Expand[p[n, x]], {n, 0, 6}]
%t A192352 reductionRules = {x^y_?EvenQ -> q[x]^(y/2),
%t A192352    x^y_?OddQ -> x q[x]^((y - 1)/2)};
%t A192352 t = Table[Last[Most[FixedPointList[Expand[#1 /. reductionRules] &, p[n, x]]]], {n, 0, 30}]
%t A192352 Table[Coefficient[Part[t, n], x, 0], {n, 1, 30}]
%t A192352   (* A192352 *)
%t A192352 Table[Coefficient[Part[t, n], x, 1], {n, 1, 30}]
%t A192352   (* A049602 *)
%Y A192352 Cf. A192232, A049602.
%K A192352 nonn
%O A192352 1,3
%A A192352 _Clark Kimberling_, Jun 29 2011