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A192462 Constant term of the reduction by x^2->x+1 of the polynomial p(n,x) defined below in Comments. 2
1, 1, 3, 18, 134, 1219, 13051, 160877, 2243285, 34910810, 599778960, 11274872675, 230192376755, 5072160696515, 119969157163845, 3031681775228370, 81517508176185730, 2323785190405594685, 70003126753631869325, 2222084456557049981875 (list; graph; refs; listen; history; text; internal format)
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 A095776

Adjacent sequences:  A192459 A192460 A192461 * A192463 A192464 A192465

KEYWORD

nonn

AUTHOR

Clark Kimberling, Jul 01 2011

STATUS

approved

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Last modified September 26 05:07 EDT 2021. Contains 347664 sequences. (Running on oeis4.)