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Constant term of the reduction by x^2->x+1 of the polynomial p(n,x)=(x+2)(x+4)...(x+2n).
2

%I #8 May 08 2014 15:18:42

%S 1,2,9,61,546,6043,79475,1209160,20873685,402896615,8595041400,

%T 200773098515,5095839723205,139624739872970,4107177047046645,

%U 129087781738773385,4316962772836390050,153048896045632212175,5733602882337419294975

%N Constant term of the reduction by x^2->x+1 of the polynomial p(n,x)=(x+2)(x+4)...(x+2n).

%C For an introduction to reductions of polynomials by substitutions such as x^2->x+1, see A192232.

%F Conjecture: a(n) +(-4*n+1)*a(n-1) +(4*n^2-6*n+1)*a(n-2)=0. - _R. J. Mathar_, May 08 2014

%e The first four polynomials p(n,x) and their reductions are as follows:

%e p(0,x)=1

%e p(1,x)=x+2 -> x+2

%e p(2,x)=(x+2)(x+4) -> 9+7x

%e p(3,x)=(x+2)(x+4)(x+6) -> 61+58x

%e From these, read

%e A192939=(1,2,9,61,...) and A192940=(0,1,7,58,...)

%t q = x^2; s = x + 1; z = 26;

%t p[0, x] := 1;

%t p[n_, x_] := (x + 2 n)*p[n - 1, x];

%t Table[Expand[p[n, x]], {n, 0, 7}]

%t reduce[{p1_, q_, s_, x_}] :=

%t FixedPoint[(s PolynomialQuotient @@ #1 +

%t PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]

%t t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];

%t u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}]

%t (* A192939 *)

%t u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}]

%t (* A192940 *)

%Y Cf. A192232, A192744, A192940.

%K nonn

%O 0,2

%A _Clark Kimberling_, Jul 13 2011