A209573 Triangle of coefficients of polynomials u(n,x) jointly generated with A209574; see the Formula section.

1, 1, 1, 2, 4, 1, 3, 9, 9, 1, 4, 17, 29, 16, 1, 5, 28, 69, 74, 25, 1, 6, 42, 138, 224, 160, 36, 1, 7, 59, 245, 541, 613, 307, 49, 1, 8, 79, 399, 1127, 1781, 1469, 539, 64, 1, 9, 102, 609, 2111, 4331, 5103, 3171, 884, 81, 1, 10, 128, 884, 3649, 9281, 14419

Offset: 1

Author

Clark Kimberling, Mar 11 2012

Keywords

• nonn
• tabl

Comments

For n>1, let r(n,k) be the k-th number in row n. Then

r(n,1)=n-1, r(n,n-1)=(n-1)^2, and r(n,n)=1. For a

discussion and guide to related arrays, see A208510.

Examples

			First five rows:
1
1...1
2...4....1
3...9....9....1
4...17...29...16...1
First three polynomials v(n,x): 1, 1 + x, 2 + 4x + x^2.
		

Crossrefs

Cf. A209574, A208510.

Programs

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n - 1, x] + v[n - 1, x];
v[n_, x_] := 2 x*u[n - 1, x] + (x + 1)*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[%]   (* A209573 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]   (* A209574 *)

Formula

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

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

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