OFFSET
1,2
COMMENTS
FORMULA
u(n,x)=u(n-1,x)+v(n-1,x)+1,
v(n,x)=x*u(n-1,x)+v(n-1,x)+1,
where u(1,x)=1, v(1,x)=1.
Also: writing T(n,m) for the general term,
T(n,1)=n for n>=1;
T(n,k)=C(n+1,2k-1) for 1<=k<=floor[(n+2)/2].
EXAMPLE
First eight rows:
1
2...1
3...4
4...10...1
5...20...6
6...35...21....1
7...56...56....8
8...84...126...36...1
First five polynomials v(n,x):
1
2 + x
3 + 4x.
4 + 10x + x^2
5 + 20x + 6x^2.
MATHEMATICA
u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + v[n - 1, x] + 1;
v[n_, x_] := x*u[n - 1, 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[%] (* A210039 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%] (* A210040 *)
Table[u[n, x] /. x -> 1, {n, 1, z}] (* A000225 *)
Table[v[n, x] /. x -> 1, {n, 1, z}] (* A000225 *)
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Clark Kimberling, Mar 17 2012
STATUS
approved