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A192373
Constant term in the reduction of the polynomial p(n,x) defined at A162517 and below in Comments.
3
1, 0, 7, 8, 77, 192, 1043, 3472, 15529, 57792, 240655, 934808, 3789653, 14963328, 60048443, 238578976, 953755537, 3798340224, 15162325975, 60438310184, 241126038941, 961476161856, 3835121918243, 15294304429744, 61000836720313, 243280700771904
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
The first five polynomials p(n,x) and their reductions are as follows:
p(0,x)=1 -> 1
p(1,x)=2x -> 2x
p(2,x)=4+x+3x^2 -> 7+4x
p(3,x)=16x+4x^2+4x^3 -> 8+28x
p(4,x)=16+8x+41x^2+10x^3+5x^4 -> 77+84x.
From these, read A192352=(1,0,7,8,77,...) and A049602=(0,2,4,28,84,...).
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