A192457 Constant term of the reduction by x^2->x+2 of the polynomial p(n,x) defined below in Comments.

0, 2, 14, 118, 1210, 14730, 208110, 3350550, 60580170, 1215657450, 26813382750, 644830644150, 16793095369050, 470839138619850, 14140985865756750, 452938463797569750, 15412288335824630250, 555226177657611710250, 21111260070730770690750

Offset: 0

Author

Clark Kimberling, Jul 01 2011

Keywords

• nonn

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.

Examples

			The first four polynomials p(n,x) and their reductions are as follows:
p(0,x)=x -> x
p(1,x)=x(2+x) -> 2+3x
p(2,x)=x(2+x)(4+x) -> 14+17x
p(3,x)=x(2+x)(4+x)(6+x) -> 118+133x.
From these, read
A192457=(0,2,14,118,...) and A192459=(1,3,17,133,...)
		

Crossrefs

Cf. A192232, A192459, A192458.

Programs

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 *)

Formula

a(n) = (2/3)*(2^n(n+1)! - (2n-1)!!). - Vaclav Potocek, Feb 04 2016