A133567 A007318 * A133566.

1, 1, 1, 1, 3, 1, 1, 6, 3, 1, 1, 10, 6, 5, 1, 1, 15, 10, 15, 5, 1, 1, 21, 15, 35, 15, 7, 1, 1, 28, 21, 70, 35, 28, 7, 1, 1, 36, 28, 126, 70, 84, 28, 9, 1, 1, 45, 36, 210, 126, 210, 84, 45, 9, 1

Offset: 1

Author

Gary W. Adamson, Sep 16 2007

Keywords

• nonn
• tabl

Comments

Row sums = A083329: (1, 2, 5, 11, 23, 47, 95, ...).

From Clark Kimberling, Feb 28 2012: (Start)

A133567 is jointly generated with A133084 as an array of coefficients of polynomials v(n,x): initially, u(1,x) = v(1,x) = 1; for n > 1, u(n,x) = u(n-1,x) + (x+1)*v(n-1)x and v(n,x) = x*u(n-1,x) + v(n-1,x) + 1. See the Mathematica section. (End)

Examples

			First few rows of the triangle:
  1;
  1,  1;
  1,  3,  1;
  1,  6,  3,  1;
  1, 10,  6,  5,  1;
  1, 15, 10, 15,  5,  1;
  1, 21, 15, 35, 15,  7,  1;
  ...
		

Crossrefs

Cf. A133566, A083329, A133084.

Programs

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];
v[n_, x_] := x*u[n - 1, x] + v[n - 1, x] + 1;
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]  (* A133567 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]  (* A133084 *)
(* Clark Kimberling, Feb 28 2012 *)

Formula

Binomial transform of triangle A133566.