A209569 Triangle of coefficients of polynomials u(n,x) jointly generated with A209570; see the Formula section.

1, 1, 1, 2, 3, 1, 3, 6, 5, 1, 4, 11, 14, 7, 1, 5, 18, 31, 26, 9, 1, 6, 27, 60, 71, 42, 11, 1, 7, 38, 105, 162, 139, 62, 13, 1, 8, 51, 170, 327, 372, 243, 86, 15, 1, 9, 66, 259, 602, 861, 754, 391, 114, 17, 1, 10, 83, 376, 1031, 1790, 1987, 1388, 591, 146, 19, 1

Offset: 1

Author

Clark Kimberling, Mar 10 2012

Keywords

• nonn
• tabl

Comments

For n>1, row n begins with n and ends with 2.

Alternating row sums: 1,0,0,1,1,0,0,1,1,0,0,1,1,...

For a discussion and guide to related arrays, see A208510.

Examples

			First five rows:
1
1...1
2...3....1
3...6....5....1
4...11...14...7...1
First three polynomials v(n,x): 1, 1 + x, 2 + 3x + x^2.
		

Crossrefs

Cf. A209570, A208510.

Programs

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n - 1, x] + v[n - 1, x];
v[n_, x_] := 2 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[%]   (* A209569 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]   (* A209570 *)

Formula

u(n,x)=x*u(n-1,x)+v(n-1,x),

v(n,x)=2x*u(n-1,x)+v(n-1,x)+1,

where u(1,x)=1, v(1,x)=1.