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A249100
Triangular array read by rows: row n gives the coefficients of the polynomial p(n,x) defined in Comments.
2
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
KEYWORD
nonn,tabl,easy
AUTHOR
Clark Kimberling, Oct 21 2014
STATUS
approved