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A192460 Constant term of the reduction by x^2->x+1 of the polynomial p(n,x) defined below in Comments. 2
0, 2, 13, 123, 1487, 21871, 378942, 7557722, 170519635, 4293742365, 119359055585, 3630473717035, 119930672906880, 4275825418586810, 163638018718726915, 6690920298998362845, 291099044600505086165, 13426830426820884360265 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

The polynomial p(n,x) is defined by recursively by p(n,x)=(nx+n-1)*p(n-1,x) with p[0,x]=1.  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..18.

EXAMPLE

The first four polynomials p(n,x) and their reductions are as follows:

p(0,x)=x -> x

p(1,x)=x(1+2x) -> 2+3x

p(2,x)=x(1+2x)(2+3x) -> 13+21x

p(3,x)=x(1+2x)(2+3x)(3+4x) -> 123+199x.

From these, read

A192460=(1,2,13,123,...) and A192461=(1,3,21,199,...)

MATHEMATICA

q[x_] := x + 1; p[0, x_] := 1;

p[n_, x_] := (n*x + n - 1)*p[n - 1, x] /; n > 0

Table[Simplify[p[n, x]], {n, 1, 5}]

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

   x^y_?OddQ -> x q[x]^((y - 1)/2)};

t = Table[FixedPoint[Expand[#1 /. reductionRules] &, p[n, x]], {n, 1, 30}]

Table[Coefficient[Part[t, n], x, 0], {n, 1, 16}]

  (* A192460 *)

Table[Coefficient[Part[t, n], x, 1], {n, 1, 16}]

  (* A192461 *)

CROSSREFS

Cf. A192232, A192461.

Sequence in context: A215715 A292437 A317196 * A004122 A086630 A151361

Adjacent sequences:  A192457 A192458 A192459 * A192461 A192462 A192463

KEYWORD

nonn

AUTHOR

Clark Kimberling, Jul 01 2011

STATUS

approved

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Last modified October 15 10:57 EDT 2018. Contains 316222 sequences. (Running on oeis4.)