cp's OEIS Frontend

A192756 Coefficient of x in the reduction by x^2->x+1 of the polynomial p(n,x) defined below in Comments.

%I A192756 #8 May 04 2014 17:26:56
%S A192756 0,1,6,17,38,75,138,243,416,699,1160,1909,3124,5093,8282,13445,21802,
%T A192756 35327,57214,92631,149940,242671,392716,635497,1028328,1663945,
%U A192756 2692398,4356473,7049006,11405619,18454770,29860539,48315464,78176163
%N A192756 Coefficient of x in the reduction by x^2->x+1 of the polynomial p(n,x) defined below in Comments.
%C A192756 The titular polynomial is defined recursively by p(n,x)=x*(n-1,x)+5n for n>0, where p(0,x)=1.  For discussions of polynomial reduction, see A192232 and A192744.
%F A192756 Conjecture: G.f.: -x*(1+3*x+x^2) / ( (x^2+x-1)*(x-1)^2 ). a(n) = A001924(n)+3*A001924(n-1)+A001924(n-2). Partial sums of A166863. - _R. J. Mathar_, May 04 2014
%t A192756 q = x^2; s = x + 1; z = 40;
%t A192756 p[0, n_] := 1; p[n_, x_] := x*p[n - 1, x] + 5 n;
%t A192756 Table[Expand[p[n, x]], {n, 0, 7}]
%t A192756 reduce[{p1_, q_, s_, x_}] :=
%t A192756 FixedPoint[(s PolynomialQuotient @@ #1 +
%t A192756        PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]
%t A192756 t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];
%t A192756 u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}]
%t A192756 (* A166863 *)
%t A192756 u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}]
%t A192756 (* A192756 *)
%Y A192756 Cf. A192744, A192232.
%K A192756 nonn
%O A192756 0,3
%A A192756 _Clark Kimberling_, Jul 09 2011