A209566 Triangle of coefficients of polynomials v(n,x) jointly generated with A209565; see the Formula section.

1, 1, 3, 1, 3, 7, 1, 3, 10, 15, 1, 3, 10, 31, 31, 1, 3, 10, 34, 88, 63, 1, 3, 10, 34, 113, 233, 127, 1, 3, 10, 34, 116, 359, 586, 255, 1, 3, 10, 34, 116, 393, 1084, 1419, 511, 1, 3, 10, 34, 116, 396, 1306, 3120, 3340, 1023, 1, 3, 10, 34, 116, 396, 1349, 4216

Offset: 1

Author

Clark Kimberling, Mar 10 2012

Keywords

• nonn
• tabl

Comments

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

Examples

			First five rows:
1
1...1
1...4...1
1...4...11...1
1...4...14...26...1
First three polynomials v(n,x): 1, 1 + x , 1 + 4x + x^2 .
		

Crossrefs

Cf. A209565, 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] + 2 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[%]   (* A209565 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]   (* A209566 *)

Formula

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

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

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