A210602 Triangle of coefficients of polynomials v(n,x) jointly generated with A210597; see the Formula section.

1, 3, 1, 6, 5, 2, 11, 14, 11, 4, 19, 34, 36, 24, 8, 32, 74, 101, 89, 52, 16, 53, 152, 251, 279, 214, 112, 32, 87, 299, 582, 769, 735, 504, 240, 64, 142, 571, 1279, 1961, 2208, 1872, 1168, 512, 128, 231, 1066, 2704, 4706, 6096, 6057, 4648, 2672, 1088

Offset: 1

Author

Clark Kimberling, Mar 24 2012

Keywords

• nonn
• tabl

Comments

For n>1, row n starts with F(n+3)-2, where F=A000045

(Fibonacci numbers), and ends with 2^(n-2).

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

Examples

			First five rows:
1
3....1
6....5....2
11...14...11...4
19...34...36...24...8
First three polynomials v(n,x): 1, 3 + x, 6 + 5x + 2x^2
		

Crossrefs

Cf. A210597, A208510.

Programs

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := 2 x*u[n - 1, x] + v[n - 1, x] + 1;
v[n_, x_] := (x + 1)*u[n - 1, 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[%]    (* A210597 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A210602 *)

Formula

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

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

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