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

Offset

0,2

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.

Links

Clark Kimberling, Rows 0..100, flattened

Formula

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

Example

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;

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 *)

Crossrefs

Cf. A249101, A245136, A087299.

Sequence in context: A135224 A209578 A268829 * A356255 A152203 A340526

Adjacent sequences: A249097 A249098 A249099 * A249101 A249102 A249103

Keyword

nonn,tabl,easy

Author

Clark Kimberling, Oct 21 2014

Status

approved