A210560 Triangle of coefficients of polynomials v(n,x) jointly generated with A210559; see the Formula section.

1, 3, 1, 5, 4, 2, 7, 9, 9, 3, 9, 16, 23, 16, 5, 11, 25, 46, 48, 30, 8, 13, 36, 80, 110, 101, 54, 13, 15, 49, 127, 215, 257, 203, 97, 21, 17, 64, 189, 378, 552, 570, 401, 172, 34, 19, 81, 268, 616, 1057, 1337, 1228, 776, 303, 55, 21, 100, 366, 948, 1862, 2772

Offset: 1

Author

Clark Kimberling, Mar 22 2012

Keywords

• nonn
• tabl

Comments

Column 1: odd positive integers (A005408)

Column 2: squares (A000290)

Row n ends with F(n), where F=A000045 (Fibonacci numbers)

Row sums: A005409

Alternating row sums: 1,2,3,4,5,6,7,8,...(A000027)

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

Examples

			First five rows:
1
3...1
5...4...2
7...9...9...3
9...16...23...16...5
First three polynomials v(n,x): 1, 3 + x , 5 + 4x + 2x^2.
		

Crossrefs

Cf. A210559, A208510.

Programs

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n - 1, x] + 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[%]  (* A210559 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]  (* A210560 *)

Formula

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