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A192924 Constant term in the reduction by (x^2->x+1) of the polynomial p(n,x) defined below at Comments.

%I A192924 #11 Jul 31 2019 14:09:28
%S A192924 1,0,1,4,20,115,761,5723,48353,454233,4701724,53204955,653749199,
%T A192924 8670930456,123500484305,1880367585200,30481594476514,524197712831867,
%U A192924 9532792177527307,182792169717039937,3686148742978363201,77989408383978583425
%N A192924 Constant term in the reduction by (x^2->x+1) of the polynomial p(n,x) defined below at Comments.
%C A192924 The titular polynomial is defined by p(n,x)=n*p(n-1,x)+(x^2)*p(n-2,x), with p(0,x)=1, p(1,x)=x.  For discussions of polynomial reduction, see A192232, A192744, and A192872.
%F A192924 Conjecture: a(n) +2*(-n+1)*a(n-1) +(n^2-3*n-1)*a(n-2) +3*(n-2)*a(n-3) +a(n-4)=0. - _R. J. Mathar_, May 08 2014
%t A192924 q = x^2; s = x + 1; z = 22;
%t A192924 p[0, x_] := 1; p[1, x_] := x;
%t A192924 p[n_, x_] := n*p[n - 1, x] + p[n - 2, x]*x^2;
%t A192924 Table[Expand[p[n, x]], {n, 0, 7}]
%t A192924 reduce[{p1_, q_, s_, x_}] :=
%t A192924 FixedPoint[(s PolynomialQuotient @@ #1 +
%t A192924        PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]
%t A192924 t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];
%t A192924 u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}]
%t A192924   (* A192924 *)
%t A192924 u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}]
%t A192924   (* A192925 *)
%Y A192924 Cf. A192232, A192744, A192925.
%K A192924 nonn
%O A192924 0,4
%A A192924 _Clark Kimberling_, Jul 12 2011