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A192755 Coefficient of x in the reduction by x^2->x+1 of the polynomial p(n,x) defined below in Comments.

%I A192755 #17 Jun 24 2017 01:02:13
%S A192755 0,1,7,19,42,82,150,263,449,753,1248,2052,3356,5469,8891,14431,23398,
%T A192755 37910,61394,99395,160885,260381,421372,681864,1103352,1785337,
%U A192755 2888815,4674283,7563234,12237658,19801038,32038847,51840041,83879049
%N A192755 Coefficient of x in the reduction by x^2->x+1 of the polynomial p(n,x) defined below in Comments.
%C A192755 The titular polynomial is defined recursively by p(n,x)=x*(n-1,x)+5n+1 for n>0, where p(0,x)=1. For discussions of polynomial reduction, see A192232 and A192744.
%F A192755 From _R. J. Mathar_, May 04 2014: (Start)
%F A192755 Conjecture: G.f.: -x*(1+4*x) / ( (x^2+x-1)*(x-1)^2 ).
%F A192755 a(n) = A001924(n)+4*A001924(n-1).
%F A192755 Partial sums of A192754. (End)
%t A192755 p[0, n_] := 1; p[n_, x_] := x*p[n - 1, x] + 5 n + 1;
%t A192755 Table[Expand[p[n, x]], {n, 0, 7}]
%t A192755 reduce[{p1_, q_, s_, x_}] :=
%t A192755 FixedPoint[(s PolynomialQuotient @@ #1 +
%t A192755        PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]
%t A192755 t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];
%t A192755 u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}]
%t A192755   (* A192754 *)
%t A192755 u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}]
%t A192755   (* A192755 *)
%Y A192755 Cf. A192754, A192744, A192232.
%K A192755 nonn
%O A192755 0,3
%A A192755 _Clark Kimberling_, Jul 09 2011