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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 March 22 18:49 EDT 2023. Contains 361433 sequences. (Running on oeis4.)