A192924 Constant term in the reduction by (x^2->x+1) of the polynomial p(n,x) defined below at Comments.

1, 0, 1, 4, 20, 115, 761, 5723, 48353, 454233, 4701724, 53204955, 653749199, 8670930456, 123500484305, 1880367585200, 30481594476514, 524197712831867, 9532792177527307, 182792169717039937, 3686148742978363201, 77989408383978583425

Offset: 0

Author

Clark Kimberling, Jul 12 2011

Keywords

• nonn

Comments

The titular polynomial is defined by p(n,x)=n*p(n-1,x)+(x^2)*p(n-2,x), with p(0,x)=1, p(1,x)=x. For discussions of polynomial reduction, see A192232, A192744, and A192872.

Crossrefs

Cf. A192232, A192744, A192925.

Programs

Mathematica

q = x^2; s = x + 1; z = 22;
p[0, x_] := 1; p[1, x_] := x;
p[n_, x_] := n*p[n - 1, x] + p[n - 2, x]*x^2;
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}]
  (* A192924 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}]
  (* A192925 *)

Formula

Conjecture: a(n) +2*(-n+1)*a(n-1) +(n^2-3*n-1)*a(n-2) +3*(n-2)*a(n-3) +a(n-4)=0. - R. J. Mathar, May 08 2014