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Coefficient of x in the reduction by x^2->x+1 of the polynomial p(n,x) defined below in Comments.
3

%I #17 Jun 24 2017 01:02:13

%S 0,1,7,19,42,82,150,263,449,753,1248,2052,3356,5469,8891,14431,23398,

%T 37910,61394,99395,160885,260381,421372,681864,1103352,1785337,

%U 2888815,4674283,7563234,12237658,19801038,32038847,51840041,83879049

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

%C 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 From _R. J. Mathar_, May 04 2014: (Start)

%F Conjecture: G.f.: -x*(1+4*x) / ( (x^2+x-1)*(x-1)^2 ).

%F a(n) = A001924(n)+4*A001924(n-1).

%F Partial sums of A192754. (End)

%t p[0, n_] := 1; p[n_, x_] := x*p[n - 1, x] + 5 n + 1;

%t Table[Expand[p[n, x]], {n, 0, 7}]

%t reduce[{p1_, q_, s_, x_}] :=

%t FixedPoint[(s PolynomialQuotient @@ #1 +

%t PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]

%t t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];

%t u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}]

%t (* A192754 *)

%t u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}]

%t (* A192755 *)

%Y Cf. A192754, A192744, A192232.

%K nonn

%O 0,3

%A _Clark Kimberling_, Jul 09 2011