A247376 Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.

1, 2, 2, 3, 5, 5, 15, 8, 8, 35, 33, 13, 80, 131, 48, 21, 171, 409, 279, 34, 355, 1180, 1375, 384, 55, 715, 3128, 5335, 2895, 89, 1410, 7858, 18510, 17029, 3840, 144, 2730, 18851, 58253, 78609, 35685, 233, 5208, 43629, 171059, 317758, 243873, 46080, 377, 9810

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 + (2*x + 1)/f(n-1,x), where f(0,x) = 1.

(Sum of numbers in row n) = A059480(n+1) for n >= 0.

(Column 1) is essentially A000045 (Fibonacci numbers).

Links

• Clark Kimberling, Rows 0..100, flattened

Crossrefs

Cf. A059480, A000045.

Programs

Mathematica

z = 15; f[x_, n_] := 1 + (2 x + 1)/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}]] (* A247376 array *)
Flatten[CoefficientList[u, x]] (* A247376 sequence *)

PARI

rown(n) = if (n==0, 1, 1 + (2*x+1)/rown(n-1));
tabl(nn) = for (n=0, nn, print(Vecrev(numerator(rown(n))))); \\ Michel Marcus, Oct 28 2014

Formula

f(0,x) = 1/1, so that p(0,x) = 1

f(1,x) = (2 + 2 x)/1, so that p(1,x) = 2 + 2 x;

f(2,x) = (3 + 5 x)/(2 + 2 x), so that p(2,x) = 3 + 5 x.

First 6 rows of the triangle of coefficients:

1

2 2

3 5

5 15 8

8 35 33

13 80 131 48