A209767 Triangle of coefficients of polynomials u(n,x) jointly generated with A209768; see the Formula section.

1, 1, 2, 2, 6, 5, 3, 12, 20, 12, 4, 21, 52, 63, 29, 5, 33, 109, 199, 187, 70, 6, 48, 200, 490, 700, 536, 169, 7, 66, 334, 1032, 1988, 2322, 1498, 408, 8, 87, 520, 1948, 4742, 7488, 7378, 4109, 985, 9, 111, 767, 3388, 10004, 19992, 26664, 22685, 11109

Offset: 1

Author

Clark Kimberling, Mar 15 2012

Keywords

• nonn
• tabl

Comments

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

Examples

			First five rows:
1
1...2
2...6....5
3...12...20...12
4...21...52...63...29
First three polynomials u(n,x): 1, 1 + 2x, 2 + 6x + 5x^2.
		

Crossrefs

Cf. A209768, 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_] := 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[%]    (* A209767 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A209768 *)

Formula

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