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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

%I

%S 1,2,2,9,12,5,64,112,68,16,625,1375,1125,420,65,7776,20736,21600,

%T 11124,2910,326,117649,369754,470596,311787,114611,22652,1957,2097152,

%U 7602176,11468800,9342976,4455424,1254976,196872,13700,43046721,176969853,309298662

%N 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.

%C 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).

%C 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)!.

%H Clark Kimberling, <a href="/A248665/b248665.txt">Table of n, a(n) for n = 1..5000</a>

%e The first six polynomials:

%e p(1,x) = 1

%e p(2,x) = 2 (x + 1)

%e p(3,x) = 9x^2 + 12 x + 5

%e p(4,x) = 4 (16 x^3 + 28 x^2 + 17 x + 4)

%e p(5,x) = 5 (125 x^4 + 275 x^3 + 225 x^2 + 84 x + 13)

%e p(6,x) = 2 (3888 x^5 + 10368 x^4 + 10800 x^3 + 5562 x^2 + 1455 x + 163)

%e First six rows of the triangle:

%e 1

%e 2 2

%e 9 12 5

%e 64 112 68 16

%e 625 1375 1125 420 65

%e 7776 20736 21600 11124 2910 326

%t t[x_, n_, k_] := t[x, n, k] = Product[n*x + n - i, {i, 1, k}];

%t p[x_, n_] := Sum[t[x, n, k], {k, 0, n - 1}];

%t TableForm[Table[Factor[p[x, n]], {n, 1, 6}]]

%t c[n_] := c[n] = Reverse[CoefficientList[p[x, n], x]];

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

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

%t u = Table[Apply[GCD, c[n]], {n, 1, 60}] (*A248666*)

%t Flatten[Position[u, 1]] (*A248667*)

%t Table[Apply[Plus, c[n]], {n, 1, 60}] (*A248668*)

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

%K nonn,tabl,easy

%O 1,2

%A _Clark Kimberling_, Oct 11 2014

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Last modified May 11 06:28 EDT 2021. Contains 343784 sequences. (Running on oeis4.)