A193004 Constant term of the reduction by x^2->x+1 of the polynomial p(n,x) defined at Comments.

1, 1, 9, 29, 75, 165, 331, 623, 1123, 1963, 3357, 5651, 9405, 15525, 25477, 41633, 67831, 110281, 179031, 290339, 470511, 762111, 1234009, 1997639, 3233305, 5232745, 8468001, 13702853, 22173123, 35878413, 58054147, 93935351, 151992475

Offset

0,3

Comments

The titular polynomials are defined recursively: p(n,x)=x*p(n-1,x)+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.: (x^4-3*x^3+10*x^2-3*x+1) / ((x-1)^3*(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^3;
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}] (* A193004 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A193005 *)

Prog

(PARI) Vec((x^4-3*x^3+10*x^2-3*x+1)/((x-1)^3*(x^2+x-1)) + O(x^100)) \\ Colin Barker, May 12 2014

Crossrefs

Cf. A192232, A192744, A192951, A193005.

Sequence in context: A273306 A318742 A024922 * A198645 A045862 A035075

Adjacent sequences: A193001 A193002 A193003 * A193005 A193006 A193007

Keyword

nonn,easy

Author

Clark Kimberling, Jul 14 2011

Status

approved