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%I #4 Mar 30 2012 18:57:34
%S 0,2,13,123,1487,21871,378942,7557722,170519635,4293742365,
%T 119359055585,3630473717035,119930672906880,4275825418586810,
%U 163638018718726915,6690920298998362845,291099044600505086165,13426830426820884360265
%N Constant term of the reduction by x^2->x+1 of the polynomial p(n,x) defined below in Comments.
%C The polynomial p(n,x) is defined by recursively by p(n,x)=(nx+n-1)*p(n-1,x) with p[0,x]=1. For an introduction to reductions of polynomials by substitutions such as x^2->x+2, see A192232.
%e The first four polynomials p(n,x) and their reductions are as follows:
%e p(0,x)=x -> x
%e p(1,x)=x(1+2x) -> 2+3x
%e p(2,x)=x(1+2x)(2+3x) -> 13+21x
%e p(3,x)=x(1+2x)(2+3x)(3+4x) -> 123+199x.
%e From these, read
%e A192460=(1,2,13,123,...) and A192461=(1,3,21,199,...)
%t q[x_] := x + 1; p[0, x_] := 1;
%t p[n_, x_] := (n*x + n - 1)*p[n - 1, x] /; n > 0
%t Table[Simplify[p[n, x]], {n, 1, 5}]
%t reductionRules = {x^y_?EvenQ -> q[x]^(y/2),
%t x^y_?OddQ -> x q[x]^((y - 1)/2)};
%t t = Table[FixedPoint[Expand[#1 /. reductionRules] &, p[n, x]], {n, 1, 30}]
%t Table[Coefficient[Part[t, n], x, 0], {n, 1, 16}]
%t (* A192460 *)
%t Table[Coefficient[Part[t, n], x, 1], {n, 1, 16}]
%t (* A192461 *)
%Y Cf. A192232, A192461.
%K nonn
%O 1,2
%A _Clark Kimberling_, Jul 01 2011
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