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

Offset: 0

Author

Clark Kimberling, Jul 15 2011

Keywords

• nonn
• easy

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

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

Crossrefs

Cf. A192232, A192744, A192951, A193049, A193046.

Programs

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]
t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];
u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}] (* A193048 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A193049 *)

PARI

Vec((x^2-x+1)*(x^4-5*x^3+9*x^2-5*x+1)/((x-1)^5*(x^2+x-1)) + O(x^100)) \\ Colin Barker, May 12 2014

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