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

0,4

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.

Links

Table of n, a(n) for n=0..21.

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

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 *)

Crossrefs

Cf. A192232, A192744, A192925.

Sequence in context: A361245 A100034 A341919 * A258664 A231539 A106567

Adjacent sequences: A192921 A192922 A192923 * A192925 A192926 A192927

Keyword

nonn

Author

Clark Kimberling, Jul 12 2011

Status

approved