A210563 Triangle of coefficients of polynomials u(n,x) jointly generated with A210564; see the Formula section.

1, 1, 2, 1, 3, 5, 1, 3, 10, 13, 1, 3, 11, 32, 34, 1, 3, 11, 40, 99, 89, 1, 3, 11, 41, 141, 299, 233, 1, 3, 11, 41, 152, 482, 887, 610, 1, 3, 11, 41, 153, 556, 1604, 2595, 1597, 1, 3, 11, 41, 153, 570, 1998, 5217, 7508, 4181, 1, 3, 11, 41, 153, 571, 2113, 7042

Offset: 1

Author

Clark Kimberling, Mar 23 2012

Keywords

• nonn
• tabl

Comments

Last terms in rows: odd-indexed Fibonacci numbers

Limiting row: A001835

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

Examples

			First five rows:
1
1...2
1...3...5
1...3...10...13
1...3...11...32...34
First three polynomials u(n,x): 1, 1 + 2x, 1 + 3x + 5x^2.
		

Crossrefs

Cf. A210564, 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] + 2 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[%]  (* A210563 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]  (* A210564 *)

Formula

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

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

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