A208516 Triangle of coefficients of polynomials u(n,x) jointly generated with A208517; see the Formula section.

1, 1, 1, 1, 2, 3, 1, 3, 6, 7, 1, 4, 9, 15, 17, 1, 5, 12, 24, 39, 41, 1, 6, 15, 34, 66, 100, 99, 1, 7, 18, 45, 98, 178, 256, 239, 1, 8, 21, 57, 135, 276, 478, 653, 577, 1, 9, 24, 70, 177, 395, 772, 1275, 1661, 1393, 1, 10, 27, 84, 224, 536, 1145, 2139, 3383, 4214

Offset: 1

Author

Clark Kimberling, Feb 28 2012

Keywords

• nonn
• tabl

Examples

			First five rows:
1
1...1
1...2...3
1...3...6...7
1...4...9...15...17
First five polynomials u(n,x):
1
1 + x
1 + 2x + 3x^2
1 + 3x + 6x^2 + 7x^3
1 + 4x + 9x^2 + 15x^3 + 17x^4
		

Crossrefs

Cf. A208517.

Programs

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];
v[n_, x_] := x*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[%]  (* A208516 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]  (* A208517 *)

Formula

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

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

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