A193046 Constant term of the reduction by x^2->x+1 of the polynomial p(n,x) defined at Comments.

1, 1, 17, 83, 275, 727, 1673, 3505, 6873, 12843, 23155, 40639, 69889, 118353, 198097, 328659, 541667, 888311, 1451433, 2365089, 3846201, 6245771, 10131747, 16423103, 26606785, 43088737, 69761873, 112925075, 182770163, 295787863

Offset

0,3

Comments

The titular polynomials are defined recursively: p(n,x)=x*p(n-1,x)+n^4, 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..29.

Index entries for linear recurrences with constant coefficients, signature (5,-9,6,1,-3,1).

Formula

a(n) = 5*a(n-1)-9*a(n-2)+6*a(n-3)+a(n-4)-3*a(n-5)+a(n-6).

G.f.: (x^5-6*x^4-x^3-21*x^2+4*x-1) / ((x-1)^4*(x^2+x-1)). - Colin Barker, May 11 2014

Mathematica

q = x^2; s = x + 1; z = 40;
p[0, x] := 1;
p[n_, x_] := x*p[n - 1, x] + n^4;
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}]   (* A193046 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}]   (* A193047 *)

Crossrefs

Cf. A192232, A192744, A192951, A193047.

Sequence in context: A142059 A343498 A318743 * A158528 A197346 A213436

Adjacent sequences: A193043 A193044 A193045 * A193047 A193048 A193049

Keyword

nonn,easy

Author

Clark Kimberling, Jul 15 2011

Status

approved