OFFSET
1,3
COMMENTS
The polynomials are given by p(n,x)=((x+d)^n-(x-d)^n)/(2d), where d=sqrt(x+4).
For an introduction to reductions of polynomials by substitutions such as x^2->x+1, see A192232.
FORMULA
Conjecture: a(n) = 2*a(n-1)+10*a(n-2)-6*a(n-3)-9*a(n-4). G.f.: -x*(x+1)*(3*x-1) / (9*x^4+6*x^3-10*x^2-2*x+1). - Colin Barker, May 09 2014
EXAMPLE
MATHEMATICA
q[x_] := x + 1; d = Sqrt[x + 4];
p[n_, x_] := ((x + d)^n - (x - d)^n )/(2 d) (* A162517 *)
Table[Expand[p[n, x]], {n, 1, 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, 1, 30}]
Table[Coefficient[Part[t, n], x, 0], {n, 1, 30}] (* A192373 *)
Table[Coefficient[Part[t, n], x, 1], {n, 1, 30}] (* A192374 *)
Table[Coefficient[Part[t, n]/2, x, 1], {n, 1, 30}] (* A192375 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Clark Kimberling, Jun 29 2011
STATUS
approved