OFFSET
1,5
COMMENTS
FORMULA
u(n,x)=x*u(n-1,x)+v(n-1,x)-1,
v(n,x)=x*u(n-1,x)+x*v(n-1,x)+1,
where u(1,x)=1, v(1,x)=1.
Also, u(n,x)=2x*u(n-1,x)+(x-x^2)*u(n-2,x)+x, where u(2,x)=x.
EXAMPLE
First six rows:
1
0...1
0...2...1
0...1...5...1
0...1...4...9....1
0...1...3...12...14...1
First three polynomials u(n,x): 1, x, 2x + x^2.
MATHEMATICA
u[1, x_] := 1; v[1, x_] := 1; z = 14;
u[n_, x_] := x*u[n - 1, x] + v[n - 1, x] - 1;
v[n_, x_] := x*u[n - 1, x] + 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[%] (* A210872 *)
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%] (* A210873 *)
Table[u[n, x] /. x -> 1, {n, 1, z}] (* A000225 *)
Table[v[n, x] /. x -> 1, {n, 1, z}] (* A083318 *)
Table[u[n, x] /. x -> -1, {n, 1, z}] (* -A077973 *)
Table[v[n, x] /. x -> -1, {n, 1, z}] (* A137470 *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Mar 29 2012
STATUS
approved