A248809 Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.

1, 2, 1, 3, 2, 1, 8, 7, 2, 1, 15, 18, 12, 2, 1, 48, 57, 30, 18, 2, 1, 105, 174, 141, 44, 25, 2, 1, 384, 561, 414, 285, 60, 33, 2, 1, 945, 1950, 1830, 810, 510, 78, 42, 2, 1, 3840, 6555, 6090, 4680, 1410, 840, 98, 52, 2, 1, 10395, 25290, 26685, 15000, 10290

Offset

0,2

Comments

The polynomial p(n,x) is the numerator of the rational function given by f(n,x) = x + (n + 1)/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 A006882 (double factorials).

Links

Clark Kimberling, Rows 0..100, flattened.

Example

f(0,x) = 1/1, so that p(0,x) = 1.
f(1,x) = (2 + x)/1, so that p(1,x) = 2 + x.
f(2,x) = (3 + 2 x + x^2)/(2 + x), so that p(2,x) = 3 + 2 x + x^2.
First 6 rows of the triangle of coefficients:
1
2    1
3    2     1
8    7     2    1
15   18   12    2    1
48   57   30    18   2    1

Mathematica

z = 15; f[x_, n_] := x + (n + 1)/f[x, n - 1]; f[x_, 0] = 1;
t = Table[Factor[f[x, n]], {n, 0, z}]
u = Numerator[t]
TableForm[Table[CoefficientList[u[[n]], x], {n, 1, z}]] (*A248809 array*)
Flatten[CoefficientList[u, x]] (*A249809 sequence*)

Prog

(PARI) rown(n) = if (n==0, 1, x + (n+1)/rown(n-1));
tabl(nn) = for (n=0, nn, print(Vecrev(numerator(rown(n))))); \\ Michel Marcus, Oct 25 2014

Crossrefs

Cf. A000982, A006882.

Sequence in context: A107880 A102228 A141675 * A021473 A266756 A035181

Adjacent sequences: A248806 A248807 A248808 * A248810 A248811 A248812

Keyword

nonn,tabl,easy

Author

Clark Kimberling, Oct 23 2014

Status

approved