A208610 Triangle of coefficients of polynomials u(n,x) jointly generated with A208611; see the Formula section.

1, 1, 1, 1, 3, 3, 1, 5, 9, 7, 1, 7, 17, 25, 17, 1, 9, 27, 55, 69, 41, 1, 11, 39, 99, 171, 187, 99, 1, 13, 53, 159, 341, 515, 501, 239, 1, 15, 69, 237, 599, 1125, 1515, 1329, 577, 1, 17, 87, 335, 967, 2141, 3591, 4373, 3497, 1393, 1, 19, 107, 455, 1469, 3713

Offset: 1

Author

Clark Kimberling, Mar 01 2012

Keywords

• nonn
• tabl

Examples

			First five rows:
1
1...1
1...3...3
1...5...9...7
1...7...17...25...17
First five polynomials u(n,x):
1
1 + x
1 + 3x + 3x^2
1 + 5x + 9x^2 + 7x^3
1 + 7x + 17x^2 + 25x^3 + 17x^4
		

Crossrefs

Cf. A208611.

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 + 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[%]  (* A208610 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]  (* A208611 *)

Formula

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

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

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