A209568 Triangle of coefficients of polynomials v(n,x) jointly generated with A209567; see the Formula section.

1, 2, 2, 3, 5, 3, 4, 10, 11, 4, 5, 17, 28, 21, 5, 6, 26, 58, 67, 36, 6, 7, 37, 105, 166, 142, 57, 7, 8, 50, 173, 350, 416, 274, 85, 8, 9, 65, 266, 659, 1011, 940, 491, 121, 9, 10, 82, 388, 1141, 2156, 2612, 1955, 829, 166, 10, 11, 101, 543, 1852, 4172, 6265

Offset: 1

Author

Clark Kimberling, Mar 10 2012

Keywords

• nonn
• tabl

Comments

Row n begins and ends with n.

Column 2: (n-1)^2, for n>1.

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

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

Examples

			First five rows:
1
2...2
3...5...3
4...10...11...4
5...17...28...21...5
First three polynomials v(n,x): 1, 2+2x, 3+5x+3x^2.
		

Crossrefs

Cf. A209567, 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_] := x*u[n - 1, x] + (x + 1)*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[%]   (* A209567 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]   (* A209568 *)

Formula

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

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

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