A248666 - OEIS

%I #5 Oct 17 2014 23:20:17

%S 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 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 4,65,2,1,4,1,10,1,4,1,74,5,4,1,26,1,20,1

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

%C 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 The first six polynomials are shown here.  The number just to the right of "=" is the GCD of the coefficients.

%e p(1,x) = 1*1

%e p(2,x) = 2*(x + 1)

%e p(3,x) = 1*(9x^2 + 12 x +  5)

%e p(4,x) = 4*(16 x^3 + 28 x^2 + 17 x + 4)

%e p(5,x) = 5*(125 x^4 + 275 x^3 + 225 x^2 + 84 x + 13)

%e 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 t[x_, n_, k_] := t[x, n, k] = Product[n*x + n - i, {i, 1, k}];

%t p[x_, n_] := Sum[t[x, n, k], {k, 0, n - 1}];

%t TableForm[Table[Factor[p[x, n]], {n, 1, 6}]]

%t c[n_] := c[n] = CoefficientList[p[x, n], x];

%t TableForm[Table[c[n], {n, 1, 10}]]   (* A248664 array *)

%t Table[Apply[GCD, c[n]], {n, 1, 60}]  (* A248666 *)

%Y Cf. A248664, A248665, A248667, A248668, A248669.

%K nonn,easy

%O 1,2

%A _Clark Kimberling_, Oct 11 2014