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A248666 Greatest common divisor of the coefficients of the polynomial p(n,x) defined in Comments.

%I A248666 #5 Oct 17 2014 23:20:17
%S A248666 1,2,1,4,5,2,1,4,1,10,1,4,13,2,5,4,1,2,1,20,1,2,1,4,5,26,1,4,1,10,1,4,
%T A248666 1,2,5,4,37,2,13,20,1,2,1,4,5,2,1,4,1,10,1,52,1,2,5,4,1,2,1,20,1,2,1,
%U A248666 4,65,2,1,4,1,10,1,4,1,74,5,4,1,26,1,20,1
%N A248666 Greatest common divisor of the coefficients of the polynomial p(n,x) defined in Comments.
%C A248666 The polynomial p(n,x) is defined as the numerator when the sum 1 + 1/(n*x + 1) + 1/((n*x + 1)(n*x + 2)) + ... + 1/((n*x + 1)(n*x + 2)...(n*x + n - 1)) is written as a fraction with denominator (n*x + 1)(n*x + 2)...(n*x + n - 1).  For more, see A248664.  For n such that the coefficients of p(n,x) are relatively prime, see A248667.
%e A248666 The first six polynomials are shown here.  The number just to the right of "=" is the GCD of the coefficients.
%e A248666 p(1,x) = 1*1
%e A248666 p(2,x) = 2*(x + 1)
%e A248666 p(3,x) = 1*(9x^2 + 12 x +  5)
%e A248666 p(4,x) = 4*(16 x^3 + 28 x^2 + 17 x + 4)
%e A248666 p(5,x) = 5*(125 x^4 + 275 x^3 + 225 x^2 + 84 x + 13)
%e A248666 p(6,x) = 2*(3888 x^5 + 10368 x^4 + 10800 x^3 + 5562 x^2 + 1455 x + 163), so that A248666 = (1,2,1,4,5,2, ...).
%t A248666 t[x_, n_, k_] := t[x, n, k] = Product[n*x + n - i, {i, 1, k}];
%t A248666 p[x_, n_] := Sum[t[x, n, k], {k, 0, n - 1}];
%t A248666 TableForm[Table[Factor[p[x, n]], {n, 1, 6}]]
%t A248666 c[n_] := c[n] = CoefficientList[p[x, n], x];
%t A248666 TableForm[Table[c[n], {n, 1, 10}]]   (* A248664 array *)
%t A248666 Table[Apply[GCD, c[n]], {n, 1, 60}]  (* A248666 *)
%Y A248666 Cf. A248664, A248665, A248667, A248668, A248669.
%K A248666 nonn,easy
%O A248666 1,2
%A A248666 _Clark Kimberling_, Oct 11 2014