|
|
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
|
|
|
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}]
Table[Coefficient[Part[t, n], x, 1], {n, 1, 20}]
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|