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

%I A248664 #13 Jul 23 2025 11:46:43
%S A248664 1,2,2,5,12,9,16,68,112,64,65,420,1125,1375,625,326,2910,11124,21600,
%T A248664 20736,7776,1957,22652,114611,311787,470596,369754,117649,13700,
%U A248664 196872,1254976,4455424,9342976,11468800,7602176,2097152,109601,1895148,14699961,65045025
%N A248664 Triangular array of coefficients of polynomials p(n,k) defined in Comments.
%C A248664 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 A248664 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 A248664 Clark Kimberling, <a href="/A248664/b248664.txt">Table of n, a(n) for n = 1..5000</a>
%e A248664 The first six polynomials:
%e A248664 p(1,x) = 1
%e A248664 p(2,x) = 2 (1 + x)
%e A248664 p(3,x) = 5 + 12 x + 9x^2
%e A248664 p(4,x) = 4 (4 + 17 x + 28 x^2 + 16 x^3)
%e A248664 p(5,x) = 5 (13 + 84 x + 225 x^2 + 275 x^3 + 125 x^4)
%e A248664 p(6,x) = 2 (163 + 1455 x + 5562 x^2 + 10800 x^3 + 10368 x^4 + 3888 x^5)
%e A248664 First six rows of the triangle:
%e A248664 1
%e A248664 2     2
%e A248664 5     12     9
%e A248664 16    68    112    64
%e A248664 65    420   1125   1375    625
%e A248664 326   2910  11124  21600   20736   7776
%t A248664 t[x_, n_, k_] := t[x, n, k] = Product[n*x + n - i, {i, 1, k}];
%t A248664 p[x_, n_] := Sum[t[x, n, k], {k, 0, n - 1}];
%t A248664 TableForm[Table[Factor[p[x, n]], {n, 1, 6}]]
%t A248664 c[n_] := c[n] = CoefficientList[p[x, n], x];
%t A248664 TableForm[Table[c[n], {n, 1, 10}]]  (* A248664 array *)
%t A248664 Flatten[Table[c[n], {n, 1, 10}]] (* A248664 sequence *)
%t A248664 u = Table[Apply[GCD, c[n]], {n, 1, 60}] (* A248666 *)
%t A248664 Flatten[Position[u, 1]]  (* A248667 *)
%t A248664 Table[Apply[Plus, c[n]], {n, 1, 60}]    (* A248668 *)
%t A248664 Table[p[x, n] /. x -> -1, {n, 1, 30}] (* A153229 signed *)
%Y A248664 Cf. A248665, A248666, A248667, A248668, A248669.
%K A248664 nonn,tabl,easy
%O A248664 1,2
%A A248664 _Clark Kimberling_, Oct 11 2014