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 A192379 Constant term of the reduction by x^2->x+1 of the polynomial p(n,x) defined below in Comments. 3
 1, 0, 5, 8, 45, 128, 505, 1680, 6089, 21120, 74909, 262680, 926485, 3258112, 11474865, 40382752, 142171985, 500432640, 1761656821, 6201182760, 21829269181, 76841888640, 270495370025, 952182350768, 3351823875225, 11798909226368 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS The polynomial p(n,x) is defined by ((x+d)^n-(x-d)^n)/(2d), where d=sqrt(x+2).  For an introduction to reductions of polynomials by substitutions such as x^2->x+1, see A192232. LINKS FORMULA Conjecture: a(n) = 2*a(n-1)+6*a(n-2)-2*a(n-3)-a(n-4). G.f.: -x*(x^2+2*x-1) / (x^4+2*x^3-6*x^2-2*x+1). - Colin Barker, May 11 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)=2+x+3x^2 -> 5+4x p(3,x)=8x+4x^2+4x^3 -> 8+20x p(4,x)=4+4x+21x^2+10x^3+5x^4 -> 45+60x. From these, read A192379=(1,0,5,8,45,...) and A192380=(0,2,4,20,60,...). MATHEMATICA q[x_] := x + 1; d = Sqrt[x + 2]; p[n_, x_] := ((x + d)^n - (x - d)^n )/(2 d)   (* Cf. 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}]   (* A192379 *) Table[Coefficient[Part[t, n], x, 1], {n, 1, 30}]   (* A192380 *) Table[Coefficient[Part[t, n]/2, x, 1], {n, 1, 30}]   (* A192381 *) CROSSREFS Cf. A192232, A192380, A192381. Sequence in context: A256401 A109292 A275571 * A117474 A294665 A323139 Adjacent sequences:  A192376 A192377 A192378 * A192380 A192381 A192382 KEYWORD nonn AUTHOR Clark Kimberling, Jun 29 2011 STATUS approved

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Last modified November 25 09:29 EST 2020. Contains 338623 sequences. (Running on oeis4.)