A249060 -id:A249060 - cp's OEIS frontend

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

Clark Kimberling, Oct 20 2014

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).

			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

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

Cf. A249059, A249060.

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 *)

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

A249059 Row sums of the triangular array at A249057.

Original entry on oeis.org

1, 5, 10, 40, 110, 430, 1420, 5720, 21340, 89980, 367400, 1627120, 7138120, 33172040, 154520080, 751616800, 3687498320, 18719834320, 96157299040, 507993654080, 2719611532000, 14911459229920, 82901747529920, 470599687507840, 2708946870815680

Offset: 0

Clark Kimberling, Oct 20 2014

			First 3 rows from A249057:
1
4    1
5    4    1,
so that a(0) = 1, a(1) = 5, a(2) = 10.

Clark Kimberling, Table of n, a(n) for n = 0..100

Cf. A249057, A249060.

z = 30; p[x_, n_] := x + (n + 2)/p[x, n - 1]; p[x_, 1] = 1;
t = Table[Factor[p[x, n]], {n, 1, z}];
u = Numerator[t];
v1 = Flatten[CoefficientList[u, x]]; (* A249057  *)
v2 = u /. x -> 1  (* A249059 *)
v3 = u /. x -> 0  (* A249060 *)

cp's OEIS Frontend

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

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