A210216 Triangle of coefficients of polynomials v(n,x) jointly generated with A210215; see the Formula section.

1, 1, 2, 1, 3, 3, 1, 3, 7, 4, 1, 3, 8, 14, 5, 1, 3, 8, 20, 25, 6, 1, 3, 8, 21, 46, 41, 7, 1, 3, 8, 21, 54, 97, 63, 8, 1, 3, 8, 21, 55, 133, 189, 92, 9, 1, 3, 8, 21, 55, 143, 309, 344, 129, 10, 1, 3, 8, 21, 55, 144, 364, 674, 591, 175, 11, 1, 3, 8, 21, 55, 144, 376, 894

Offset: 1

Author

Clark Kimberling, Mar 19 2012

Keywords

• nonn
• tabl

Comments

Limiting row: even-indexed Fibonacci numbers, A001906.

n-th row sum: -1+2*n

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

Examples

			First five rows:
1
1...2
1...3...3
1...3...7...4
1...3...8...14...5
First three polynomials v(n,x): 1, 1 + 2x , 1 + 3x + 3x^2.
		

Crossrefs

Cf. A210215, 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] + 1;
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[%]    (* A210215 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A210216 *)
Table[u[n, x] /. x -> 1, {n, 1, z}]  (* A000225 *)
Table[v[n, x] /. x -> 1, {n, 1, z}]  (* A000225 *)
Table[u[n, x] /. x -> -1, {n, 1, z}] (* A137470 *)
Table[v[n, x] /. x -> -1, {n, 1, z}] (* A137470 *)

Formula

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

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

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