A210233 Triangle of coefficients of polynomials u(n,x) jointly generated with A210234; see the Formula section.

1, 2, 1, 3, 5, 1, 4, 10, 12, 1, 5, 18, 30, 27, 1, 6, 27, 68, 83, 58, 1, 7, 39, 119, 225, 217, 121, 1, 8, 52, 200, 454, 680, 544, 248, 1, 9, 68, 300, 866, 1566, 1928, 1320, 503, 1, 10, 85, 440, 1450, 3332, 5014, 5216, 3121, 1014, 1, 11, 105, 605, 2350, 6182

Offset: 1

Author

Clark Kimberling, Mar 20 2012

Keywords

• nonn
• tabl

Comments

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

Examples

			First five rows:
1
2...1
3...5....1
4...10...12...1
5...18...30...27...1
First three polynomials u(n,x): 1, 2 + x, 3 + 5x + x^2.
		

Crossrefs

Cf. A210234, 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_] := (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[%]      (* A210233 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]      (* A210234 *)

Formula

u(n,x)=x*u(n-1,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.