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A193048 Constant term of the reduction by x^2->x+1 of the polynomial p(n,x) defined at Comments.

%I A193048 #14 Feb 18 2025 08:17:57
%S A193048 1,0,1,2,8,25,68,163,357,730,1417,2642,4774,8417,14556,24793,41729,
%T A193048 69582,115187,189614,310786,507715,827356,1345697,2185703,3546350,
%U A193048 5749603,9316428,15089782,24433615,39554862,64024437,103620219,167691032
%N A193048 Constant term of the reduction by x^2->x+1 of the polynomial p(n,x) defined at Comments.
%C A193048 The titular polynomials are defined recursively:  p(n,x)=x*p(n-1,x)+n(4-5*n^2+n^4)/120, 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 A193048 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (6,-14,15,-5,-4,4,-1).
%F A193048 a(n) = 6*a(n-1)-14*a(n-2)+15*a(n-3)-5*a(n-4)-4*a(n-5)+4*a(n-6)-a(n-7).
%F A193048 G.f.: (x^2-x+1)*(x^4-5*x^3+9*x^2-5*x+1) / ((x-1)^5*(x^2+x-1)). - _Colin Barker_, May 12 2014
%t A193048 q = x^2; s = x + 1; z = 40;
%t A193048 p[0, x] := 1;
%t A193048 p[n_, x_] := x*p[n - 1, x] + n (-1 + n^2) (-4 + n^2)/120;
%t A193048 Table[Expand[p[n, x]], {n, 0, 7}]
%t A193048 reduce[{p1_, q_, s_, x_}] := FixedPoint[(s PolynomialQuotient @@ #1 + PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]
%t A193048 t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];
%t A193048 u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}] (* A193048 *)
%t A193048 u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A193049 *)
%o A193048 (PARI) Vec((x^2-x+1)*(x^4-5*x^3+9*x^2-5*x+1)/((x-1)^5*(x^2+x-1)) + O(x^100)) \\ _Colin Barker_, May 12 2014
%Y A193048 Cf. A192232, A192744, A192951, A193049, A193046.
%K A193048 nonn,easy
%O A193048 0,4
%A A193048 _Clark Kimberling_, Jul 15 2011