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A193008 Constant term of the reduction by x^2->x+1 of the polynomial p(n,x) defined at Comments.

%I A193008 #15 Feb 18 2025 08:17:47
%S A193008 1,2,10,31,78,170,339,636,1144,1997,3412,5740,9549,15758,25854,42243,
%T A193008 68818,111878,181615,294520,477276,773057,1251720,2026296,3279673,
%U A193008 5307770,8589394,13899271,22490934,36392642,58886187,95281620,154170784
%N A193008 Constant term of the reduction by x^2->x+1 of the polynomial p(n,x) defined at Comments.
%C A193008 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 A193008 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (4,-5,1,2,-1).
%F A193008 a(n) = 4*a(n-1)-5*a(n-2)+a(n-3)+2*a(n-4)-a(n-5).
%F A193008 G.f.: (7*x^2-2*x+1)/((x-1)^3*(x^2+x-1)). [_Colin Barker_, Nov 12 2012]
%t A193008 q = x^2; s = x + 1; z = 40;
%t A193008 p[0, x] := 1;
%t A193008 p[n_, x_] := x*p[n - 1, x] + n^3 + 1;
%t A193008 Table[Expand[p[n, x]], {n, 0, 7}]
%t A193008 reduce[{p1_, q_, s_, x_}] :=
%t A193008 FixedPoint[(s PolynomialQuotient @@ #1 +
%t A193008        PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]
%t A193008 t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];
%t A193008 u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}] (* A193008 *)
%t A193008 u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A193009 *)
%Y A193008 Cf. A192232, A192744, A192951, A193009.
%K A193008 nonn,easy
%O A193008 0,2
%A A193008 _Clark Kimberling_, Jul 14 2011