|
%I #10 Jul 01 2016 23:58:44
%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
|
