OFFSET
0,2
COMMENTS
The polynomial p(n,x) is defined by recursively by p(n,x)=(x+2n)*p(n-1,x) with p[0,x]=x. For an introduction to reductions of polynomials by substitutions such as x^2->x+2, see A192232.
FORMULA
a(n) = (2/3)*(2^n(n+1)! - (2n-1)!!). - Vaclav Potocek, Feb 04 2016
EXAMPLE
MATHEMATICA
q[x_] := x + 2; p[0, x_] := x;
p[n_, x_] := (x + 2 n)*p[n - 1, x] /; n > 0
Table[Simplify[p[n, x]], {n, 0, 5}]
reductionRules = {x^y_?EvenQ -> q[x]^(y/2),
x^y_?OddQ -> x q[x]^((y - 1)/2)};
t = Table[FixedPoint[Expand[#1 /. reductionRules] &, p[n, x]], {n, 0, 30}]
Table[Coefficient[Part[t, n], x, 0], {n, 1, 16}] (* A192457 *)
Table[Coefficient[Part[t, n]/2, x, 0], {n, 1, 16}] (* A192458 *)
Table[Coefficient[Part[t, n], x, 1], {n, 1, 16}] (* A192459 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Clark Kimberling, Jul 01 2011
STATUS
approved