A192350 Constant term of the reduction (by x^2->x+1) of polynomial p(n,x) identified in Comments.
1, 0, 6, 4, 64, 128, 896, 2752, 14208, 52224, 238592, 946176, 4110336, 16830464, 71598080, 297140224, 1253048320, 5229707264, 21973303296, 91924463616, 385642135552, 1614916091904, 6770569248768, 28364203098112, 118885634277376
Offset: 1
Keywords
Examples
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)=5+x+x^2 -> 6+2x p(3,x)=15x+3x^2+x^3 -> 4+20x. From these, we read A192350=(1,0,6,4,...) and A192351=(0,1,2,20...)
Programs
-
Mathematica
q[x_] := x + 1; d = Sqrt[x + 5]; 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}] (* A192350 *) Table[Coefficient[Part[t, n], x, 1], {n, 1, 30}] (* A192351 *)
Formula
Conjecture: a(n) = 2*a(n-1)+12*a(n-2)-8*a(n-3)-16*a(n-4). G.f.: -x*(6*x^2+2*x-1) / (16*x^4+8*x^3-12*x^2-2*x+1). [Colin Barker, Jan 17 2013]
Comments