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%I #7 Mar 30 2012 18:57:34
%S 1,1,3,18,134,1219,13051,160877,2243285,34910810,599778960,
%T 11274872675,230192376755,5072160696515,119969157163845,
%U 3031681775228370,81517508176185730,2323785190405594685,70003126753631869325,2222084456557049981875
%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+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) = 1.
%e p(1,x)=1+x -> 1+x.
%e p(2,x)=(1+x)(1+2x) -> 3+5x.
%e p(3,x)=(1+x)(1+2x)(1+3x) -> 18+29x.
%e p(4,x)=(1+x)(1+2x)(1+3x)(1+4x) -> 134+217x.
%e From these, read
%e A192462=(1,1,3,18,134,...) and A192463=(0,1,5,29,217,...)
%t q[x_] := x + 1; p[0, x_] := 1;
%t p[n_, x_] := (n*x + 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, 20}]
%t Table[Coefficient[Part[t, n], x, 0], {n, 1, 20}]
%t (* A192462 *)
%t Table[Coefficient[Part[t, n], x, 1], {n, 1, 20}]
%t (* A192463 *)
%Y Cf. A192232, A192463.
%K nonn
%O 0,3
%A _Clark Kimberling_, Jul 01 2011
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