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A192352
Constant term of the reduction of the polynomial p(n,x)=(1/2)((x+1)^n+(x-1)^n) by x^2->x+1.
5
1, 0, 2, 1, 9, 13, 51, 106, 322, 771, 2135, 5401, 14445, 37324, 98514, 256621, 673933, 1760997, 4615823, 12075526, 31628466, 82781215, 216761547, 567428401, 1485645049, 3889310328, 10182603746, 26657986681, 69792188337, 182717232061
OFFSET
1,3
COMMENTS
1
FORMULA
Empirical G.f.: -x*(x^3-x^2-2*x+1)/((x^2-3*x+1)*(x^2-x-1)). [Colin Barker, Sep 11 2012]
EXAMPLE
For an introduction to reductions of polynomials by substitutions such as x^2->x+1, see A192232.
The first six polynomials p(n,x) and their reductions are as follows:
p(0,x)=1 -> 1
p(1,x)=x -> x
p(2,x)=1+x^2 -> 2+x
p(3,x)=3x+x^3 -> 1+5x
p(4,x)=1+6x^2+x^4 -> 9+9x
p(5,x)=5x+10x^3+x^5 -> 13+30x.
From these, we read
A192352=(1,0,2,1,9,13...) and A049602=(0,1,1,5,9,30...).
MATHEMATICA
q[x_] := x + 1; d = 1;
p[n_, x_] := ((x + d)^n + (x - d)^n )/
2 (* similar to polynomials defined at A161516 *)
Table[Expand[p[n, x]], {n, 0, 6}]
reductionRules = {x^y_?EvenQ -> q[x]^(y/2),
x^y_?OddQ -> x q[x]^((y - 1)/2)};
t = Table[Last[Most[FixedPointList[Expand[#1 /. reductionRules] &, p[n, x]]]], {n, 0, 30}]
Table[Coefficient[Part[t, n], x, 0], {n, 1, 30}]
(* A192352 *)
Table[Coefficient[Part[t, n], x, 1], {n, 1, 30}]
(* A049602 *)
CROSSREFS
Sequence in context: A099599 A085488 A072265 * A180001 A204371 A199887
KEYWORD
nonn
AUTHOR
Clark Kimberling, Jun 29 2011
STATUS
approved