A210223 Triangle of coefficients of polynomials u(n,x) jointly generated with A210224; see the Formula section.

1, 2, 1, 2, 5, 1, 2, 7, 10, 1, 2, 7, 22, 17, 1, 2, 7, 26, 57, 26, 1, 2, 7, 26, 89, 126, 37, 1, 2, 7, 26, 97, 266, 247, 50, 1, 2, 7, 26, 97, 346, 695, 442, 65, 1, 2, 7, 26, 97, 362, 1127, 1618, 737, 82, 1, 2, 7, 26, 97, 362, 1319, 3298, 3425, 1162, 101, 1, 2, 7, 26

Offset: 1

Author

Clark Kimberling, Mar 20 2012

Keywords

• nonn
• tabl

Comments

Limiting row: A001075

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

Examples

			First five rows:
1
2...1
2...5...1
2...7...10...1
2...7...22...17...1
First three polynomials u(n,x): 1, 2 + x, 2 + 5x + x^2.
		

Crossrefs

Cf. A210224, 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] + 1;
v[n_, x_] := 2 x*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[%]     (* A210223 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]     (* A210224 *)

Formula

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

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

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