A249248 Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.

1, 1, 3, 1, 7, 3, 1, 12, 22, 3, 1, 18, 69, 40, 3, 1, 25, 159, 241, 61, 3, 1, 33, 310, 883, 582, 85, 3, 1, 42, 543, 2465, 3393, 1155, 112, 3, 1, 52, 882, 5798, 13805, 9786, 2032, 142, 3, 1, 63, 1354, 12110, 44253, 57521, 23368, 3294, 175, 3, 1, 75, 1989

Offset

1,3

Comments

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

Links

Clark Kimberling, Table of n, a(n) for n = 1..5050

Example

f(1,x) = 1/1, so that p(1,x) = 1
f(2,x) = (1 + 3 x)/1, so that p(2,x) = 1 + 3 x;
f(3,x) = (1 + 7 x + 3 x^2)/(1 + 3 x), so that p(3,x) = 1 + 7 x + 3 x^2.
First 6 rows of the triangle of coefficients:
1
1      3
1      7      3
1      12     22     3
1      18     69     40    3
1      25     150    241   61    3

Mathematica

z = 14; f[n_, x_] := x + 1 + n*x/f[n - 1, x]; f[1, x_] = 1;
t = Table[Factor[f[n, x]], {n, 1, z}]
u = Numerator[t]
TableForm[Table[CoefficientList[u[[n]], x], {n, 1, z}]] (*A249248 array*)
Flatten[CoefficientList[u, x]] (*A249248 sequence*)

Prog

(PARI) rown(n) = if (n==1, 1, x + 1 + n*x/rown(n-1));
tabl(nn) = for (n=1, nn, print(Vecrev(numerator(rown(n))))); \\ Michel Marcus, Oct 30 2014

Crossrefs

Cf. A249247.

Sequence in context: A133115 A210192 A249755 * A104797 A130330 A050227

Adjacent sequences: A249245 A249246 A249247 * A249249 A249250 A249251

Keyword

nonn,tabl,easy

Author

Clark Kimberling, Oct 24 2014

Status

approved