A249159 Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.

1, 1, 1, 3, 2, 2, 4, 7, 2, 2, 15, 18, 24, 4, 4, 24, 57, 30, 36, 4, 4, 105, 174, 282, 88, 100, 8, 8, 192, 561, 414, 570, 120, 132, 8, 8, 945, 1950, 3660, 1620, 2040, 312, 336, 16, 16, 1920, 6555, 6090, 9360, 2820, 3360, 392, 416, 16, 16, 10395, 25290, 53370

Offset: 0

Author

Clark Kimberling, Oct 23 2014

Keywords

• nonn
• tabl
• easy

Comments

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

(Sum of numbers in row n) = A000982(n+1) for n >= 0.

Column 1 is essentially A081405.

Links

• Clark Kimberling, Rows 0..100, flattened

Crossrefs

Cf. A000982, A081405.

Programs

Mathematica

z = 15; f[x_, n_] := 1 + n/(2 f[x, n - 1]); f[x_, 1] = 1;
t = Table[Factor[f[x, n]], {n, 1, z}]
u = Numerator[t]
TableForm[Table[CoefficientList[u[[n]], x], {n, 1, z}]] (* A249159 array *)
Flatten[CoefficientList[u, x]] (* A249159 sequence *)

Formula

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) = (3 + 2 x + x^2)/(1 + x), so that p(2,x) = 3 + 2 x + x^2.

First 6 rows of the triangle of coefficients:

1

1 1

3 2 2

4 7 2 2

15 18 24 4 4

24 57 30 36 4 4