A249057 Triangular array: Row n shows the coefficients of polynomials p(n,x) defined in Comments.

1, 4, 1, 5, 4, 1, 24, 11, 4, 1, 35, 52, 18, 4, 1, 192, 123, 84, 26, 4, 1, 315, 660, 285, 120, 35, 4, 1, 1920, 1545, 1500, 545, 160, 45, 4, 1, 3465, 9180, 4680, 2820, 930, 204, 56, 4, 1, 23040, 22005, 27180, 11220, 4740, 1470, 252, 68, 4, 1, 45045, 142380

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.

Row sums give A249059(n) for n >= 1.

First column is A249060 (n-th term = n!! for n >= 0).

Links

Clark Kimberling, Table of n, a(n) for n = 0..5049

Example

f(0,x) = 1/1, so that p(0,x) = 1
f(1,x) = (4 + x)/1, so that p(1,x) = 4 + x;
f(2,x) = (5 + 4 x + x^2)/(1 + x), so that p(2,x) = 5 + 4 x + x^2.
First 6 rows of the triangle of coefficients:
1
4    1
5    4     1
24   11    4    1
35   52    18   4    1
192  123   84   26   4   1

Mathematica

z = 12; f[x_, n_] := x + (n+3)/f[x, n - 1];
f[x_, 0] = 1; t = Table[Factor[f[x, n]], {n, 0, z}];
u = Numerator[t]; TableForm[Rest[Table[CoefficientList[u[[n]], x], {n, 0, z}]]];
Flatten[CoefficientList[u, x]] (* A249057 sequence *)

Prog

(PARI) f(n) = if (n, x + (n + 3)/f(n-1), 1);
row(n) = Vecrev(numerator(f(n)), 0); \\ Michel Marcus, Nov 25 2022

Crossrefs

Cf. A249059, A249060.

Sequence in context: A030352 A212643 A104571 * A105721 A099310 A021880

Adjacent sequences: A249054 A249055 A249056 * A249058 A249059 A249060

Keyword

nonn,tabl,easy

Author

Clark Kimberling, Oct 20 2014

Status

approved