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

%I A192457 #13 Feb 05 2016 06:47:48
%S A192457 0,2,14,118,1210,14730,208110,3350550,60580170,1215657450,26813382750,
%T A192457 644830644150,16793095369050,470839138619850,14140985865756750,
%U A192457 452938463797569750,15412288335824630250,555226177657611710250,21111260070730770690750
%N A192457 Constant term of the reduction by x^2->x+2 of the polynomial p(n,x) defined below in Comments.
%C A192457 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.
%F A192457 a(n) = (2/3)*(2^n(n+1)! - (2n-1)!!). - _Vaclav Potocek_, Feb 04 2016
%e A192457 The first four polynomials p(n,x) and their reductions are as follows:
%e A192457 p(0,x)=x -> x
%e A192457 p(1,x)=x(2+x) -> 2+3x
%e A192457 p(2,x)=x(2+x)(4+x) -> 14+17x
%e A192457 p(3,x)=x(2+x)(4+x)(6+x) -> 118+133x.
%e A192457 From these, read
%e A192457 A192457=(0,2,14,118,...) and A192459=(1,3,17,133,...)
%t A192457 q[x_] := x + 2; p[0, x_] := x;
%t A192457 p[n_, x_] := (x + 2 n)*p[n - 1, x] /; n > 0
%t A192457 Table[Simplify[p[n, x]], {n, 0, 5}]
%t A192457 reductionRules = {x^y_?EvenQ -> q[x]^(y/2),
%t A192457    x^y_?OddQ -> x q[x]^((y - 1)/2)};
%t A192457 t = Table[FixedPoint[Expand[#1 /. reductionRules] &, p[n, x]], {n, 0, 30}]
%t A192457 Table[Coefficient[Part[t, n], x, 0], {n, 1, 16}]  (* A192457 *)
%t A192457 Table[Coefficient[Part[t, n]/2, x, 0], {n, 1, 16}]  (* A192458 *)
%t A192457 Table[Coefficient[Part[t, n], x, 1], {n, 1, 16}]  (* A192459 *)
%Y A192457 Cf. A192232, A192459, A192458.
%K A192457 nonn
%O A192457 0,2
%A A192457 _Clark Kimberling_, Jul 01 2011