A249130 -id:A249130 - cp's OEIS frontend

A249131 p(1,n), where the polynomial p(n,x) is defined in Comments; sum of the numbers in row n of the triangular array at A249130.

1, 3, 5, 17, 37, 139, 361, 1473, 4361, 19091, 62701, 291793, 1044205, 5129307, 19748177, 101817089, 417787921, 2250495523, 9770678101, 54780588561, 250194150581, 1455367098923, 6959638411705, 41888448785857, 208919770666777, 1298019439099059

Offset: 0

Clark Kimberling, Oct 22 2014

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

			The first 3 rows of the array at A249130:
  1
  2    1
  2    2    1,
so that the first 3 terms of A249131 are 1,3,5.

Cf. A249130.

z = 15; p[x_, n_] := x + 2 Floor[n/2]/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}]] (* A249130 array *)
u /. x -> 1  (* A249131 *)

A249128 Triangular array: row n gives the coefficients of the polynomial p(n,x) defined in Comments.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 2, 4, 5, 1, 1, 6, 11, 7, 8, 1, 1, 6, 18, 26, 10, 11, 1, 1, 24, 50, 46, 58, 14, 15, 1, 1, 24, 96, 154, 86, 102, 18, 19, 1, 1, 120, 274, 326, 444, 156, 177, 23, 24, 1, 1, 120, 600, 1044, 756, 954, 246, 272, 28, 29, 1, 1, 720, 1764, 2556, 3708, 1692, 2016, 384, 416, 34, 35, 1, 1

Offset: 0

Clark Kimberling, Oct 22 2014

The polynomial p(n,x) is the numerator of the rational function given by f(n,x) = x + floor((n+1)/2)/f(n-1,x), where f(0,x) = 1.
(Sum of numbers in row n) = A056953(n) for n >= 0.
Column 1 consists of repeated factorials (A000142), as in A081123.

			f(0,x) = 1/1, so that p(0,x) = 1;
f(1,x) = (1 + x)/1, so that p(1,x) = 1 + x;
f(2,x) = (1 + x + x^2)/(1 + x), so that p(2,x) = 1 + x + x^2.
First 6 rows of the triangle of coefficients:
  1
  1    1
  1    1    1
  2    3    1    1
  2    4    5    1    1
  6    11   7    8    1    1

Clark Kimberling, Rows 0..100, flattened

Cf. A056953, A000142, A081123, A249130.

z = 15; p[x_, n_] := x + Floor[n/2]/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}]] (* A249128 array *)
Flatten[CoefficientList[u, x]] (* A249128 sequence *)

cp's OEIS Frontend

A249131 p(1,n), where the polynomial p(n,x) is defined in Comments; sum of the numbers in row n of the triangular array at A249130.

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