A209773 Triangle of coefficients of polynomials u(n,x) jointly generated with A209774; see the Formula section.

1, 1, 2, 2, 6, 5, 2, 11, 21, 13, 3, 17, 48, 67, 34, 3, 25, 92, 188, 206, 89, 4, 33, 154, 422, 684, 619, 233, 4, 44, 238, 809, 1756, 2365, 1829, 610, 5, 54, 348, 1411, 3801, 6833, 7882, 5334, 1597, 5, 68, 484, 2285, 7369, 16471, 25302, 25549, 15393

Offset: 1

Author

Clark Kimberling, Mar 15 2012

Keywords

• nonn
• tabl

Comments

Last term in row n: F(2n+1), where F=A000045, the Fibonacci numbers

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

Examples

			First five rows:
1
1...2
2...6....5
2...11...21...13
3...17...48...67...34
First three polynomials u(n,x): 1, 1 + 2x, 2 + 6x + 5x^2.
		

Crossrefs

Cf. A209774, A208510.

Programs

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n - 1, x] + (x + 1)*v[n - 1, x];
v[n_, x_] := (x + 1)*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[%]    (* A209773 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A209774 *)

Formula

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

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

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