cp's OEIS Frontend

A193044 Constant term of the reduction by x^2->x+1 of the polynomial p(n,x) defined at Comments.

%I A193044 #8 Jun 13 2015 00:53:55
%S A193044 1,0,2,5,13,28,56,105,189,330,564,949,1579,2606,4276,6987,11383,18506,
%T A193044 30042,48719,78951,127880,207062,335195,542533,878028,1420886,2299265,
%U A193044 3720529,6020200,9741164,15761829,25503489,41265846,66769896
%N A193044 Constant term of the reduction by x^2->x+1 of the polynomial p(n,x) defined at Comments.
%C A193044 The titular polynomials are defined recursively:  p(n,x)=x*p(n-1,x)+n(-1+n^2)/6, 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 A193044 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (4,-5,1,2,-1).
%F A193044 a(n)=4*a(n-1)-5*a(n-2)+a(n-3)+2*a(n-4)-a(n-5).
%F A193044 G.f.: ( 1+7*x^2-4*x^3+x^4-4*x ) / ( (x^2+x-1)*(x-1)^3 ). - _R. J. Mathar_, May 04 2014
%t A193044 q = x^2; s = x + 1; z = 40;
%t A193044 p[0, x] := 1;
%t A193044 p[n_, x_] := x*p[n - 1, x] + n (n^2 - 1)/6;
%t A193044 Table[Expand[p[n, x]], {n, 0, 7}]
%t A193044 reduce[{p1_, q_, s_, x_}] :=
%t A193044 FixedPoint[(s PolynomialQuotient @@ #1 +
%t A193044        PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]
%t A193044 t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];
%t A193044 u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}]
%t A193044   (* A193044 *)
%t A193044 u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}]
%t A193044   (* A193045 *)
%Y A193044 Cf. A192232, A192744, A192951, A193045, A179991 (first differences).
%K A193044 nonn,easy
%O A193044 0,3
%A A193044 _Clark Kimberling_, Jul 15 2011