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

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) = 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).

Examples

			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
		

Links

• Clark Kimberling, Rows 0..100, flattened.

Crossrefs

Cf. A000982, A006882.

Programs

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*)

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