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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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