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Constant term of the reduction by x^2->x+1 of the polynomial p(n,x) defined at Comments.
2

%I #11 Jun 13 2015 00:53:55

%S 1,2,10,31,78,170,339,636,1144,1997,3412,5740,9549,15758,25854,42243,

%T 68818,111878,181615,294520,477276,773057,1251720,2026296,3279673,

%U 5307770,8589394,13899271,22490934,36392642,58886187,95281620,154170784

%N Constant term of the reduction by x^2->x+1 of the polynomial p(n,x) defined at Comments.

%C The titular polynomials are defined recursively: p(n,x)=x*p(n-1,x)+1+n^3, with p(0,x)=1. For an introduction to reductions of polynomials by substitutions such as x^2->x+1, see A192232 and A192744.

%H <a href="/index/Rec">Index entries for linear recurrences with constant coefficients</a>, signature (4,-5,1,2,-1).

%F a(n)=4*a(n-1)-5*a(n-2)+a(n-3)+2*a(n-4)-a(n-5).

%F G.f.: (7*x^2-2*x+1)/((x-1)^3*(x^2+x-1)). [_Colin Barker_, Nov 12 2012]

%t q = x^2; s = x + 1; z = 40;

%t p[0, x] := 1;

%t p[n_, x_] := x*p[n - 1, x] + n^3 + 1;

%t Table[Expand[p[n, x]], {n, 0, 7}]

%t reduce[{p1_, q_, s_, x_}] :=

%t FixedPoint[(s PolynomialQuotient @@ #1 +

%t PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]

%t t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];

%t u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}]

%t (* A193008 *)

%t u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}]

%t (* A193009 *)

%Y Cf. A192232, A192744, A192951, A193009.

%K nonn,easy

%O 0,2

%A _Clark Kimberling_, Jul 14 2011