A209820 Triangle of coefficients of polynomials v(n,x) jointly generated with A209819; see the Formula section.

1, 2, 2, 2, 6, 5, 2, 8, 18, 12, 2, 8, 30, 52, 29, 2, 8, 34, 104, 146, 70, 2, 8, 34, 136, 342, 402, 169, 2, 8, 34, 144, 514, 1080, 1090, 408, 2, 8, 34, 144, 594, 1848, 3306, 2920, 985, 2, 8, 34, 144, 610, 2360, 6370, 9872, 7746, 2378, 2, 8, 34, 144, 610, 2552

Offset: 1

Author

Clark Kimberling, Mar 23 2012

Keywords

• nonn
• tabl

Comments

Let T(n,k) be the general term.

T(n,n): A000129

T(n,n-1): 2*A071667

Row sums: A003462

Alternating row sums: 1,0,1,0,1,0,1,0,...

Limiting row: F(3), F(6),F(9),...where F=A000045 (Fibonacci numbers)

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

Examples

			First five rows:
1
2...2
2...6...5
2...8...18...12
2...8...30...52...29
First three polynomials v(n,x): 1, 2 + 2x , 2 + 6x + 5x^2.
		

Crossrefs

Cf. A209819, A208510.

Programs

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n - 1, x] + 2 x*v[n - 1, x] + 1;
v[n_, x_] := (x + 1)*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[%]    (* A209819 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A209820 *)

Formula

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

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

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