A249128 Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.

1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 2, 4, 5, 1, 1, 6, 11, 7, 8, 1, 1, 6, 18, 26, 10, 11, 1, 1, 24, 50, 46, 58, 14, 15, 1, 1, 24, 96, 154, 86, 102, 18, 19, 1, 1, 120, 274, 326, 444, 156, 177, 23, 24, 1, 1, 120, 600, 1044, 756, 954, 246, 272, 28, 29, 1, 1, 720, 1764, 2556, 3708, 1692, 2016, 384, 416, 34, 35, 1, 1

Offset: 0

Author

Clark Kimberling, Oct 22 2014

Keywords

• nonn
• tabl
• easy

Comments

The polynomial p(n,x) is the numerator of the rational function given by f(n,x) = x + floor((n+1)/2)/f(n-1,x), where f(0,x) = 1.

(Sum of numbers in row n) = A056953(n) for n >= 0.

Column 1 consists of repeated factorials (A000142), as in A081123.

Examples

			f(0,x) = 1/1, so that p(0,x) = 1;
f(1,x) = (1 + x)/1, so that p(1,x) = 1 + x;
f(2,x) = (1 + x + x^2)/(1 + x), so that p(2,x) = 1 + x + x^2.
First 6 rows of the triangle of coefficients:
  1
  1    1
  1    1    1
  2    3    1    1
  2    4    5    1    1
  6    11   7    8    1    1
		

Links

• Clark Kimberling, Rows 0..100, flattened

Crossrefs

Cf. A056953, A000142, A081123, A249130.

Programs

Mathematica

z = 15; p[x_, n_] := x + Floor[n/2]/p[x, n - 1]; p[x_, 1] = 1;
t = Table[Factor[p[x, n]], {n, 1, z}]
u = Numerator[t]
TableForm[Table[CoefficientList[u[[n]], x], {n, 1, z}]] (* A249128 array *)
Flatten[CoefficientList[u, x]] (* A249128 sequence *)