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

1, 1, 3, 1, 5, 7, 1, 5, 17, 17, 1, 5, 21, 53, 41, 1, 5, 21, 81, 157, 99, 1, 5, 21, 89, 289, 449, 239, 1, 5, 21, 89, 361, 973, 1253, 577, 1, 5, 21, 89, 377, 1389, 3133, 3433, 1393, 1, 5, 21, 89, 377, 1565, 5085, 9745, 9273, 3363, 1, 5, 21, 89, 377, 1597, 6285

Offset: 1

Author

Clark Kimberling, Mar 23 2012

Keywords

• nonn
• tabl

Comments

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

T(n,n): A001333

T(n,n-1): A088210

Row sums: A003561

Alternating row sums: 1,-2,3,-4,5,-6,7,-8,...

Limiting row: F(2), F(5),F(8),...where F=A000045 (Fibonacci numbers)

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

Examples

			First five rows:
1
1...3
1...5...7
1...5...17...17
1...5...21...53...41
First three polynomials u(n,x): 1, 1 + 3x, 1 + 5x + 7x^2.
		

Crossrefs

Cf. A209820, 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.