A248664 Triangular array of coefficients of polynomials p(n,k) defined in Comments.

1, 2, 2, 5, 12, 9, 16, 68, 112, 64, 65, 420, 1125, 1375, 625, 326, 2910, 11124, 21600, 20736, 7776, 1957, 22652, 114611, 311787, 470596, 369754, 117649, 13700, 196872, 1254976, 4455424, 9342976, 11468800, 7602176, 2097152, 109601, 1895148, 14699961, 65045025

Offset: 1

Author

Clark Kimberling, Oct 11 2014

Keywords

• nonn
• tabl
• easy

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)!.

Examples

			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
		

Links

• Clark Kimberling, Table of n, a(n) for n = 1..5000

Crossrefs

Cf. A248665, A248666, A248667, A248668, A248669.

Programs

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 *)