A249138 Triangular array read by rows: row n gives the coefficients of the polynomial p(n,x) defined in Comments.

1, 1, 1, 2, 1, 1, 2, 4, 1, 1, 6, 5, 7, 1, 1, 6, 18, 8, 10, 1, 1, 24, 26, 46, 12, 14, 1, 1, 24, 96, 58, 86, 16, 18, 1, 1, 120, 154, 326, 118, 156, 21, 23, 1, 1, 120, 600, 444, 756, 198, 246, 26, 28, 1, 1, 720, 1044, 2556, 1152, 1692, 324, 384, 32, 34, 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+2)/2)/f(n-1,x), where f(0,x) = 1.

(Sum of numbers in row n) = A056952(n+2) for n >= 0.

(Column 1) is essentially A081123 (repeated factorials).

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) = (2 + x + x^2)/(1 + x), so that p(2,x) = 2 + x + x^2.
First 6 rows of the triangle of coefficients:
  1
  1    1
  2    1    1
  2    4    1    1
  6    5    7    1    1
  6    18   8    10   1   1
		

Links

• Clark Kimberling, Rows 0..100, flattened

Crossrefs

Cf. A056952, A081123, A249128.

Programs

Mathematica

z = 15; p[x_, n_] := x + Floor[(n+1)/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}]] (* A249138 array *)
Flatten[CoefficientList[u, x]] (* A249138 sequence *)