A192460 Constant term of the reduction by x^2->x+1 of the polynomial p(n,x) defined below in Comments.

0, 2, 13, 123, 1487, 21871, 378942, 7557722, 170519635, 4293742365, 119359055585, 3630473717035, 119930672906880, 4275825418586810, 163638018718726915, 6690920298998362845, 291099044600505086165, 13426830426820884360265

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 A359124 A086630

Adjacent sequences: A192457 A192458 A192459 * A192461 A192462 A192463

Keyword

nonn

Author

Clark Kimberling, Jul 01 2011

Status

approved