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A248665 Triangular array of coefficients of polynomials p(n,x) defined in Comments; these are the polynomials defined at A248664, but here the coefficients are written in the order of decreasing powers of x. 7
1, 2, 2, 9, 12, 5, 64, 112, 68, 16, 625, 1375, 1125, 420, 65, 7776, 20736, 21600, 11124, 2910, 326, 117649, 369754, 470596, 311787, 114611, 22652, 1957, 2097152, 7602176, 11468800, 9342976, 4455424, 1254976, 196872, 13700, 43046721, 176969853, 309298662 (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 (x + 1)

p(3,x) = 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)

First six rows of the triangle:

1

2     2

9     12     5

64    112    68     16

625   1375   1125   420    65

7776  20736  21600  11124  2910   326

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] = Reverse[CoefficientList[p[x, n], x]];

TableForm[Table[c[n], {n, 1, 10}]] (* A248665 array *)

Flatten[Table[c[n], {n, 1, 10}]]   (* A248665 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*)

CROSSREFS

Cf. A248664, A248666, A248667, A248668, A248669.

Sequence in context: A154100 A002880 A225465 * A066324 A143146 A298663

Adjacent sequences:  A248662 A248663 A248664 * A248666 A248667 A248668

KEYWORD

nonn,tabl,easy

AUTHOR

Clark Kimberling, Oct 11 2014

STATUS

approved

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Last modified April 22 19:11 EDT 2021. Contains 343177 sequences. (Running on oeis4.)