A193048 - OEIS

A193048

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

1, 0, 1, 2, 8, 25, 68, 163, 357, 730, 1417, 2642, 4774, 8417, 14556, 24793, 41729, 69582, 115187, 189614, 310786, 507715, 827356, 1345697, 2185703, 3546350, 5749603, 9316428, 15089782, 24433615, 39554862, 64024437, 103620219, 167691032

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,4

COMMENTS

The titular polynomials are defined recursively: p(n,x)=x*p(n-1,x)+n(4-5*n^2+n^4)/120, 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..33.

Index entries for linear recurrences with constant coefficients, signature (6,-14,15,-5,-4,4,-1).

FORMULA

a(n) = 6*a(n-1)-14*a(n-2)+15*a(n-3)-5*a(n-4)-4*a(n-5)+4*a(n-6)-a(n-7).

G.f.: (x^2-x+1)*(x^4-5*x^3+9*x^2-5*x+1) / ((x-1)^5*(x^2+x-1)). - Colin Barker, May 12 2014

MATHEMATICA

q = x^2; s = x + 1; z = 40;

p[0, x] := 1;

p[n_, x_] := x*p[n - 1, x] + n (-1 + n^2) (-4 + n^2)/120;

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

reduce[{p1_, q_, s_, x_}] := FixedPoint[(s PolynomialQuotient @@ #1 + PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]