A192343 -id:A192343

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A192342

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

+0
2

2, 7, 100, 28051, 2357659852, 16675673548656023155, 834234264904007920903714901139450715276, 2087840426219791385723375854976408025594408461778898567573217959566013061037427

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OFFSET

1,1

COMMENTS

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

LINKS

Table of n, a(n) for n=1..8.

EXAMPLE

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

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

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

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

From these, we read

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

MATHEMATICA

q[x_] := x + 2;

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

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

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

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

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

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

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

(* A192342 *)

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

(* A192343 *)

CROSSREFS

Cf. A192232, A192343.

KEYWORD

nonn

AUTHOR

Clark Kimberling, Jun 28 2011

STATUS

approved

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