A209754 Triangle of coefficients of polynomials v(n,x) jointly generated with A209753; see the Formula section.

1, 3, 1, 5, 6, 1, 9, 16, 10, 1, 15, 39, 38, 15, 1, 25, 84, 117, 76, 21, 1, 41, 172, 308, 286, 136, 28, 1, 67, 337, 744, 894, 612, 225, 36, 1, 109, 642, 1685, 2496, 2228, 1191, 351, 45, 1, 177, 1196, 3646, 6423, 7088, 4978, 2157, 523, 55, 1, 287, 2191

Offset: 1

Author

Clark Kimberling, Mar 14 2012

Keywords

• nonn
• tabl

Comments

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

Examples

			First five rows:
1
3....1
5....6....1
9....16...10...1
15...39...38...15...1
First three polynomials v(n,x): 1, 3 + x , 5 + 6x + x^2.
		

Crossrefs

Cf. A209653, A208510.

Programs

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n - 1, x] + (x + 1)*v[n - 1, x];
v[n_, 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[%]    (* A209753 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A209754 *)

Formula

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

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

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