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

0, 1, 6, 17, 38, 75, 138, 243, 416, 699, 1160, 1909, 3124, 5093, 8282, 13445, 21802, 35327, 57214, 92631, 149940, 242671, 392716, 635497, 1028328, 1663945, 2692398, 4356473, 7049006, 11405619, 18454770, 29860539, 48315464, 78176163

Offset

0,3

Comments

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.

Links

Table of n, a(n) for n=0..33.

Formula

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

Mathematica

q = x^2; s = x + 1; z = 40;
p[0, n_] := 1; p[n_, x_] := x*p[n - 1, x] + 5 n;
Table[Expand[p[n, x]], {n, 0, 7}]
reduce[{p1_, q_, s_, x_}] :=
FixedPoint[(s PolynomialQuotient @@ #1 +
       PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]
t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];
u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}]
(* A166863 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}]
(* A192756 *)

Crossrefs

Cf. A192744, A192232.

Sequence in context: A132127 A023621 A000385 * A004799 A085278 A366104

Adjacent sequences: A192753 A192754 A192755 * A192757 A192758 A192759

Keyword

nonn

Author

Clark Kimberling, Jul 09 2011

Status

approved