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A192348
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Constant term of the reduction (by x^2->x+1) of polynomial p(n,x) identified in Comments.
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2
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1, 0, 4, 4, 36, 88, 432, 1408, 5776, 20736, 80320, 297792, 1132096, 4242304, 16028928, 60276736, 227287296, 855703552, 3224482816, 12144337920, 45752574976, 172339107840, 649223532544, 2445572276224, 9212566081536, 34703459811328
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OFFSET
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1,3
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COMMENTS
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To define the polynomials p(n,x), let d=sqrt(x+3); then p(n,x)=(1/2)((x+d)^n+(x-d)^n). These are similar to polynomials at A161516.
For an introduction to reductions of polynomials by substitutions such as x^2->x+1, see A192232.
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LINKS
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FORMULA
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Conjecture: a(n) = 2*a(n-1)+8*a(n-2)-4*a(n-3)-4*a(n-4). G.f.: -x*(4*x^2+2*x-1) / (4*x^4+4*x^3-8*x^2-2*x+1). [Colin Barker, Jan 17 2013]
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EXAMPLE
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The first four polynomials p(n,x) and their reductions are as follows:
p(0,x)=1 -> 1
p(1,x)=x -> x
p(2,x)=3+x+x^2 -> 4+2x
p(3,x)=9x+3x^2+x^3 -> 4+14x.
From these, we read
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MATHEMATICA
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q[x_] := x + 1; d = Sqrt[x + 3];
p[n_, x_] := ((x + d)^n + (x - d)^n )/
2 (* similar to polynomials defined at A161516 *)
Table[Expand[p[n, x]], {n, 0, 4}]
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}]
Table[Coefficient[Part[t, n], x, 1], {n, 1, 30}]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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