cp's OEIS Frontend

A192313 Constant term of the reduction of n-th polynomial at A157751 by x^2->x+1.

%I A192313 #7 Nov 13 2012 12:46:02
%S A192313 1,2,5,13,34,91,247,680,1893,5319,15056,42867,122605,351898,1012729,
%T A192313 2920521,8435362,24392655,70599403,204472264
%N A192313 Constant term of the reduction of n-th polynomial at A157751 by x^2->x+1.
%C A192313 For an introduction to reductions of polynomials by substitutions such as x^2->x+1, see A192232.
%F A192313 Empirical G.f.: x*(x+1)*(x^2-3*x+1)/(x^4+6*x^3+x^2-4*x+1). [_Colin Barker_, Nov 13 2012]
%e A192313 The first five polynomials at A157751 and their reductions are as follows:
%e A192313 p0(x)=1 -> 1
%e A192313 p1(x)=2+x -> 2+x
%e A192313 p2(x)=4+2x+x^2 -> 5+3x
%e A192313 p3(x)=8+4x+4x^2+x^3 -> 13+10x
%e A192313 p4(x)=16+8x+12x^2+4x^3+x^4 -> 34+31x.
%e A192313 From these, we read
%e A192313 A192313=(1,2,5,13,34,...) and A192314=(0,1,3,19,31,...)
%t A192313 q[x_] := x + 1;
%t A192313 p[0, x_] := 1;
%t A192313 p[n_, x_] := (x + 1)*p[n - 1, x] + p[n - 1, -x] /; n > 0  (* A157751 *)
%t A192313 Table[Simplify[p[n, x]], {n, 0, 5}]
%t A192313 reductionRules = {x^y_?EvenQ -> q[x]^(y/2), x^y_?OddQ -> x q[x]^((y - 1)/2)};
%t A192313 t = Table[Last[Most[FixedPointList[Expand[#1 /. reductionRules] &, p[n, x]]]], {n, 0, 20}]
%t A192313 Table[Coefficient[Part[t, n], x, 0], {n, 1, 20}]
%t A192313   (* A192313 *)
%t A192313 Table[Coefficient[Part[t, n], x, 1], {n, 1, 20}]
%t A192313   (* A192337 *)
%Y A192313 Cf. A192232, A192337.
%K A192313 nonn
%O A192313 1,2
%A A192313 _Clark Kimberling_, Jun 28 2011