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A248664 Triangular array of coefficients of polynomials p(n,k) defined in Comments 15
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 (list; table; graph; refs; listen; history; text; internal format)
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

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

Sequence in context: A208872 A208581 A233357 * A143195 A192799 A181354

Adjacent sequences:  A248661 A248662 A248663 * A248665 A248666 A248667

KEYWORD

nonn,tabl,easy

AUTHOR

Clark Kimberling, Oct 11 2014

STATUS

approved

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Last modified March 29 15:10 EDT 2017. Contains 284273 sequences.