OFFSET
1,2
COMMENTS
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.
EXAMPLE
The first six polynomials are shown here. The number just to the right of "=" is the GCD of the coefficients.
p(1,x) = 1*1
p(2,x) = 2*(x + 1)
p(3,x) = 1*(9x^2 + 12 x + 5)
p(4,x) = 4*(16 x^3 + 28 x^2 + 17 x + 4)
p(5,x) = 5*(125 x^4 + 275 x^3 + 225 x^2 + 84 x + 13)
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, ...).
MATHEMATICA
t[x_, n_, k_] := t[x, n, k] = Product[n*x + n - i, {i, 1, k}];
p[x_, n_] := Sum[t[x, n, k], {k, 0, n - 1}];
TableForm[Table[Factor[p[x, n]], {n, 1, 6}]]
c[n_] := c[n] = CoefficientList[p[x, n], x];
TableForm[Table[c[n], {n, 1, 10}]] (* A248664 array *)
Table[Apply[GCD, c[n]], {n, 1, 60}] (* A248666 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Oct 11 2014
STATUS
approved