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

0,2

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

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

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 *)

Prog

(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

Crossrefs

Cf. A059480, A000045.

Sequence in context: A035537 A285407 A292802 * A101779 A169891 A137756

Adjacent sequences: A247373 A247374 A247375 * A247377 A247378 A247379

Keyword

nonn,tabf,easy

Author

Clark Kimberling, Oct 23 2014

Status

approved