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A193006 Constant term of the reduction by x^2->x+1 of the polynomial p(n,x) defined at Comments.

%I A193006 #13 Feb 18 2025 08:17:40
%S A193006 1,0,8,27,72,160,323,610,1102,1929,3302,5562,9261,15292,25100,41023,
%T A193006 66844,108684,176447,286158,463746,751165,1216298,1968982,3186937,
%U A193006 5157720,8346608,13506435,21855312,35364184,57222107,92589082,149814166
%N A193006 Constant term of the reduction by x^2->x+1 of the polynomial p(n,x) defined at Comments.
%C A193006 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 A193006 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (4,-5,1,2,-1).
%F A193006 a(n) = 4*a(n-1)-5*a(n-2)+a(n-3)+2*a(n-4)-a(n-5).
%F A193006 G.f.: (2*x^4-6*x^3+13*x^2-4*x+1)/((x-1)^3*(x^2+x-1)). [_Colin Barker_, Nov 12 2012]
%t A193006 q = x^2; s = x + 1; z = 40;
%t A193006 p[0, x] := 1;
%t A193006 p[n_, x_] := x*p[n - 1, x] + n^3 - 1;
%t A193006 Table[Expand[p[n, x]], {n, 0, 7}]
%t A193006 reduce[{p1_, q_, s_, x_}] :=
%t A193006 FixedPoint[(s PolynomialQuotient @@ #1 +
%t A193006        PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]
%t A193006 t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];
%t A193006 u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}]
%t A193006   (* A193006 *)
%t A193006 u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}]
%t A193006   (* A193007 *)
%Y A193006 Cf. A192232, A192744, A192951, A193007.
%K A193006 nonn,easy
%O A193006 0,3
%A A193006 _Clark Kimberling_, Jul 14 2011