A249139 Triangular array read by rows: row n gives the coefficients of the polynomial p(n,x) defined in Comments.

1, 3, 1, 5, 2, 11, 7, 1, 21, 16, 3, 43, 41, 12, 1, 85, 94, 34, 4, 171, 219, 99, 18, 1, 341, 492, 261, 60, 5, 683, 1101, 678, 195, 25, 1, 1365, 2426, 1692, 576, 95, 6, 2731, 5311, 4149, 1644, 340, 33, 1, 5461, 11528, 9959, 4488, 1106, 140, 7, 10923, 24881

Offset: 0

Author

Clark Kimberling, Oct 23 2014

Keywords

• nonn
• tabf
• easy

Comments

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

(Sum of numbers in row n) = A006130(n+1) for n >= 0.

(Column 1) is essentially A001045.

Examples

			f(0,x) = 1/1, so that p(0,x) = 1
f(1,x) = (3 + x)/1, so that p(1,x) = 3 + x;
f(2,x) = (5 + 2 x)/(3 + x), so that p(2,x) = 5 + 2 x.
First 6 rows of the triangle of coefficients:
1
3    1
5    2
11   7    1
21   16   3
43   41   12  1
		

Links

• Clark Kimberling, Rows 0..100, flattened

Crossrefs

Cf. A006130, A001045.

Programs

Mathematica

z = 15; f[x_, n_] := 1 + (x + 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}]] (* A249139 array *)
Flatten[CoefficientList[u, x]] (* A249139 sequence *)