A209151 Triangle of coefficients of polynomials u(n,x) jointly generated with A208337; see the Formula section.

1, 2, 1, 3, 4, 2, 4, 8, 8, 3, 5, 13, 19, 15, 5, 6, 19, 36, 42, 28, 8, 7, 26, 60, 91, 89, 51, 13, 8, 34, 92, 170, 216, 182, 92, 21, 9, 43, 133, 288, 446, 489, 363, 164, 34, 10, 53, 184, 455, 826, 1105, 1068, 709, 290, 55, 11, 64, 246, 682, 1414, 2219, 2619

Offset: 1

Author

Clark Kimberling, Mar 07 2012

Keywords

• nonn
• tabl

Comments

Last term in each row is a Fibonacci number (A000045).

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

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

Examples

			First five rows:
1
2...1
3...4....2
4...8....8....3
5...13...19...15...5
First three polynomials v(n,x): 1, 2 + x, 3 + 4x + 2x^2.
		

Crossrefs

Cf. A208337, A208510.

Programs

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + (x + 1)*v[n - 1, x];
v[n_, x_] := x*u[n - 1, x] + 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[%]    (* A209151 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A208337 *)

Formula

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

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

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