OFFSET
0,2
COMMENTS
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
KEYWORD
AUTHOR
Clark Kimberling, Oct 21 2014
STATUS
approved