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).
These polynomials occur in connection with factorials of numbers of the form [n/k] = floor(n/k); e.g., Sum_{n >= 0} ([n/k]!^k)/n! = Sum_{n >= 0} (n!^k)*p(k,n)/(k*n + k - 1)!.
LINKS
Clark Kimberling, Table of n, a(n) for n = 1..5000
EXAMPLE
The first six polynomials:
p(1,x) = 1
p(2,x) = 2 (1 + x)
p(3,x) = 5 + 12 x + 9x^2
p(4,x) = 4 (4 + 17 x + 28 x^2 + 16 x^3)
p(5,x) = 5 (13 + 84 x + 225 x^2 + 275 x^3 + 125 x^4)
p(6,x) = 2 (163 + 1455 x + 5562 x^2 + 10800 x^3 + 10368 x^4 + 3888 x^5)
First six rows of the triangle:
1
2 2
5 12 9
16 68 112 64
65 420 1125 1375 625
326 2910 11124 21600 20736 7776
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 *)
Flatten[Table[c[n], {n, 1, 10}]] (* A248664 sequence *)
u = Table[Apply[GCD, c[n]], {n, 1, 60}] (* A248666 *)
Flatten[Position[u, 1]] (* A248667 *)
Table[Apply[Plus, c[n]], {n, 1, 60}] (* A248668 *)
Table[p[x, n] /. x -> -1, {n, 1, 30}] (* A153229 signed *)
CROSSREFS
KEYWORD
AUTHOR
Clark Kimberling, Oct 11 2014
STATUS
approved