A193006 Constant term of the reduction by x^2->x+1 of the polynomial p(n,x) defined at Comments.

1, 0, 8, 27, 72, 160, 323, 610, 1102, 1929, 3302, 5562, 9261, 15292, 25100, 41023, 66844, 108684, 176447, 286158, 463746, 751165, 1216298, 1968982, 3186937, 5157720, 8346608, 13506435, 21855312, 35364184, 57222107, 92589082, 149814166

Offset

0,3

Comments

The titular polynomials are defined recursively: p(n,x)=x*p(n-1,x)-1+n^3, with p(0,x)=1. For an introduction to reductions of polynomials by substitutions such as x^2->x+1, see A192232 and A192744.

Links

Table of n, a(n) for n=0..32.

Index entries for linear recurrences with constant coefficients, signature (4,-5,1,2,-1).

Formula

a(n)=4*a(n-1)-5*a(n-2)+a(n-3)+2*a(n-4)-a(n-5).

G.f.: (2*x^4-6*x^3+13*x^2-4*x+1)/((x-1)^3*(x^2+x-1)). [Colin Barker, Nov 12 2012]

Mathematica

q = x^2; s = x + 1; z = 40;
p[0, x] := 1;
p[n_, x_] := x*p[n - 1, x] + n^3 - 1;
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}]
  (* A193006 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}]
  (* A193007 *)

Crossrefs

Cf. A192232, A192744, A192951, A193007.

Sequence in context: A246992 A351601 A276920 * A212745 A229423 A166867

Adjacent sequences: A193003 A193004 A193005 * A193007 A193008 A193009

Keyword

nonn,easy

Author

Clark Kimberling, Jul 14 2011

Status

approved