A193044 Constant term of the reduction by x^2->x+1 of the polynomial p(n,x) defined at Comments.

1, 0, 2, 5, 13, 28, 56, 105, 189, 330, 564, 949, 1579, 2606, 4276, 6987, 11383, 18506, 30042, 48719, 78951, 127880, 207062, 335195, 542533, 878028, 1420886, 2299265, 3720529, 6020200, 9741164, 15761829, 25503489, 41265846, 66769896

Offset

0,3

Comments

The titular polynomials are defined recursively: p(n,x)=x*p(n-1,x)+n(-1+n^2)/6, 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..34.

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.: ( 1+7*x^2-4*x^3+x^4-4*x ) / ( (x^2+x-1)*(x-1)^3 ). - R. J. Mathar, May 04 2014

Mathematica

q = x^2; s = x + 1; z = 40;
p[0, x] := 1;
p[n_, x_] := x*p[n - 1, x] + n (n^2 - 1)/6;
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}]
  (* A193044 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}]
  (* A193045 *)

Crossrefs

Cf. A192232, A192744, A192951, A193045, A179991 (first differences).

Sequence in context: A026522 A216378 A225690 * A122491 A320933 A290194

Adjacent sequences: A193041 A193042 A193043 * A193045 A193046 A193047

Keyword

nonn,easy

Author

Clark Kimberling, Jul 15 2011

Status

approved