A209153 Triangle of coefficients of polynomials u(n,x) jointly generated with A208340; see the Formula section.

1, 2, 1, 4, 5, 2, 7, 14, 11, 3, 11, 31, 38, 22, 5, 16, 60, 103, 93, 43, 8, 22, 106, 239, 298, 212, 81, 13, 29, 175, 497, 802, 782, 459, 150, 21, 37, 274, 952, 1909, 2393, 1917, 958, 273, 34, 46, 411, 1710, 4143, 6410, 6570, 4465, 1942, 491, 55, 56, 595

Offset: 1

Author

Clark Kimberling, Mar 07 2012

Keywords

• nonn
• tabl

Comments

Every row ends with a Fibonacci number (A000045).

Alternating row sums: 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
4....5....2
7....14...11...3
11...31...38...22...5
First three polynomials v(n,x): 1, 2 + x, 4 + 5x + 2x^2.
		

Crossrefs

Cf. A208340, 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 + 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[%]    (* A209153 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A208340 *)

Formula

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