A209583 Triangle of coefficients of polynomials u(n,x) jointly generated with A209584; see the Formula section.

1, 1, 1, 3, 3, 1, 5, 10, 6, 1, 9, 23, 24, 10, 1, 15, 51, 71, 49, 15, 1, 25, 104, 188, 178, 90, 21, 1, 41, 205, 452, 552, 390, 153, 28, 1, 67, 391, 1025, 1530, 1396, 776, 245, 36, 1, 109, 730, 2218, 3927, 4394, 3169, 1435, 374, 45, 1, 177, 1339, 4636, 9502

Offset: 1

Author

Clark Kimberling, Mar 11 2012

Keywords

• nonn
• tabl

Comments

Alternating row sums: 1,0,1,0,1,0,1,0,1,0,1,0,...

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

Examples

			First five rows:
1
1...1
3...3....1
5...10...6....1
9...23...24...10...1
First three polynomials u(n,x): 1, 1 + x, 3 + 3x + x^2.
		

Crossrefs

Cf. A209584, 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];
v[n_, x_] := (x + 1)*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[%]   (* A209583 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]   (* A209584 *)

Formula

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

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

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