A249100 Triangular array read by rows: row n gives the coefficients of the polynomial p(n,x) defined in Comments.

1, 3, 1, 5, 3, 1, 21, 12, 3, 1, 45, 48, 21, 3, 1, 231, 177, 81, 32, 3, 1, 585, 855, 450, 120, 45, 3, 1, 3465, 3240, 2070, 930, 165, 60, 3, 1, 9945, 18000, 10890, 4110, 1695, 216, 77, 3, 1, 65835, 71505, 57330, 28560, 7245, 2835, 273, 96, 3, 1, 208845, 443835, 300195, 143640, 64155, 11781, 4452, 336, 117, 3, 1

Offset: 0

Author

Clark Kimberling, Oct 21 2014

Keywords

• nonn
• tabl
• easy

Comments

The polynomial p(n,x) is the numerator of the rational function given by f(n,x) = x + (2*n+1)/f(n-1,x), where f(0,x) = 1.

(Sum of numbers in row n) = A249101(n) for n >= 0.

(n-th term of column 1) = A235136(n) for n >= 1.

Examples

			f(0,x) = 1/1, so that p(0,x) = 1;
f(1,x) = (3 + x)/1, so that p(1,x) = 3 + x;
f(2,x) = (5 + 3*x + x^2)/(3 + x), so that p(2,x) = 5 + 3*x + x^2.
First 6 rows of the triangle of coefficients:
    1;
    3,   1;
    5,   3,   1;
   21,  12,   3,   1;
   45,  48,  21,   3,   1;
  231, 177,  81,  32,   3,   1;
		

Links

• Clark Kimberling, Rows 0..100, flattened

Crossrefs

Cf. A249101, A245136, A087299.

Programs

Mathematica

z = 11; p[x_, n_] := x + (2 n - 1)/p[x, n - 1]; p[x_, 1] = 1;
t = Table[Factor[p[x, n]], {n, 1, z}]
u = Numerator[t]
TableForm[Table[CoefficientList[u[[n]], x], {n, 1, z}]] (* A249100 array *)
Flatten[CoefficientList[u, x]] (* A249100 sequence *)
v = u /. x -> 1  (* A249101 *)
u /. x -> 0  (* A235136 *)
T[ n_Integer, k_Integer] := (T[n, k] = If[n<2, Boole[0==k], T[n-1, k-1] + (2*n-1)*T[n-2 ,k] ]); Join @@ Table[T[n, k], {n, 10}, {k, 0, n-1}] (* Michael Somos, Oct 27 2022 *)

Formula

T(n, k) = T(n-1, k-1) + (2*n-1)*T(n-2, k). - Michael Somos, Oct 27 2022