A192342 - OEIS

%I #4 Mar 30 2012 18:57:34

%S 2,7,100,28051,2357659852,16675673548656023155,

%T 834234264904007920903714901139450715276,

%U 2087840426219791385723375854976408025594408461778898567573217959566013061037427

%N Constant term of the reduction of n-th polynomial at A158983 by x^2->x+2.

%C For an introduction to reductions of polynomials by substitutions such as x^2->x+2, see A192232.

%e The first three polynomials at A158983 and their reductions are as follows:

%e p0(x)=2+x -> 2+x

%e p1(x)=5+4x+x^2 -> 7+5x

%e p2(x)=26+40x+26x^2+8x^3+x^4 -> 100+95x.

%e From these, we read

%e A192342=(2,7,100,...) and A192343=(1,5,95,...)

%t q[x_] := x + 2;

%t p[0, x_] := x + 2;

%t p[n_, x_] := 1 + p[n - 1, x]^2 /; n > 0  (* polynomials defined at A158983 *)

%t Table[Expand[p[n, x]], {n, 0, 4}]

%t reductionRules = {x^y_?EvenQ -> q[x]^(y/2),

%t    x^y_?OddQ -> x q[x]^((y - 1)/2)};

%t t = Table[Last[Most[FixedPointList[Expand[#1 /. reductionRules] &, p[n, x]]]], {n, 0,9}]

%t Table[Coefficient[Part[t, n], x, 0], {n, 1, 9}]

%t   (* A192342 *)

%t Table[Coefficient[Part[t, n], x, 1], {n, 1, 9}]

%t   (* A192343 *)

%Y Cf. A192232, A192343.

%K nonn

%O 1,1

%A _Clark Kimberling_, Jun 28 2011