A192462 Constant term of the reduction by x^2->x+1 of the polynomial p(n,x) defined below in Comments.

1, 1, 3, 18, 134, 1219, 13051, 160877, 2243285, 34910810, 599778960, 11274872675, 230192376755, 5072160696515, 119969157163845, 3031681775228370, 81517508176185730, 2323785190405594685, 70003126753631869325, 2222084456557049981875

Offset

0,3

Comments

The polynomial p(n,x) is defined by recursively by p(n,x)=(nx+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=0..19.

Example

The first four polynomials p(n,x) and their reductions are as follows:
p(0,x) = 1.
p(1,x)=1+x -> 1+x.
p(2,x)=(1+x)(1+2x) -> 3+5x.
p(3,x)=(1+x)(1+2x)(1+3x) -> 18+29x.
p(4,x)=(1+x)(1+2x)(1+3x)(1+4x) -> 134+217x.
From these, read
A192462=(1,1,3,18,134,...) and A192463=(0,1,5,29,217,...)

Mathematica

q[x_] := x + 1; p[0, x_] := 1;
p[n_, x_] := (n*x + 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, 20}]
Table[Coefficient[Part[t, n], x, 0], {n, 1, 20}]
  (* A192462 *)
Table[Coefficient[Part[t, n], x, 1], {n, 1, 20}]
  (* A192463 *)

Crossrefs

Cf. A192232, A192463.

Sequence in context: A236342 A060909 A074545 * A168072 A251733 A355103

Adjacent sequences: A192459 A192460 A192461 * A192463 A192464 A192465

Keyword

nonn

Author

Clark Kimberling, Jul 01 2011

Status

approved