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A193004 Constant term of the reduction by x^2->x+1 of the polynomial p(n,x) defined at Comments.

%I A193004 #15 Feb 18 2025 08:17:35
%S A193004 1,1,9,29,75,165,331,623,1123,1963,3357,5651,9405,15525,25477,41633,
%T A193004 67831,110281,179031,290339,470511,762111,1234009,1997639,3233305,
%U A193004 5232745,8468001,13702853,22173123,35878413,58054147,93935351,151992475
%N A193004 Constant term of the reduction by x^2->x+1 of the polynomial p(n,x) defined at Comments.
%C A193004 The titular polynomials are defined recursively:  p(n,x)=x*p(n-1,x)+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 A193004 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (4,-5,1,2,-1).
%F A193004 a(n) = 4*a(n-1)-5*a(n-2)+a(n-3)+2*a(n-4)-a(n-5).
%F A193004 G.f.: (x^4-3*x^3+10*x^2-3*x+1) / ((x-1)^3*(x^2+x-1)). - _Colin Barker_, May 12 2014
%t A193004 q = x^2; s = x + 1; z = 40;
%t A193004 p[0, x] := 1;
%t A193004 p[n_, x_] := x*p[n - 1, x] + n^3;
%t A193004 Table[Expand[p[n, x]], {n, 0, 7}]
%t A193004 reduce[{p1_, q_, s_, x_}] :=
%t A193004 FixedPoint[(s PolynomialQuotient @@ #1 + PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]
%t A193004 t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];
%t A193004 u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}] (* A193004 *)
%t A193004 u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A193005 *)
%o A193004 (PARI) Vec((x^4-3*x^3+10*x^2-3*x+1)/((x-1)^3*(x^2+x-1)) + O(x^100)) \\ _Colin Barker_, May 12 2014
%Y A193004 Cf. A192232, A192744, A192951, A193005.
%K A193004 nonn,easy
%O A193004 0,3
%A A193004 _Clark Kimberling_, Jul 14 2011