OFFSET
1,3
COMMENTS
Row sums, u(n,1): A000129
Row sums, v(n,1): A001333
As triangle T(n,k) with 0 <= k <= n, it is (0, 2/3, 1/3, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (3, -4/3, -2/3, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Feb 27 2012
FORMULA
u(n,x) = u(n-1,x) + x*v(n-1,x),
v(n,x) = 2x*u(n-1,x) + x*v(n-1,x),
where u(1,x)=1, v(1,x)=1.
From Philippe Deléham, Feb 27 2012: (Start)
As triangle T(n,k), 0 <= k <= n:
T(n,k) = T(n-1,k) + T(n-1,k-1) - T(n-2,k-1) + 2*T(n-2,k-2) with T(0,0) = 1, T(1,0) = 0, T(1,1) = 3 and T(n,k) = 0 if k < 0 or if k > n.
G.f.: (1-(1-2*y)*x)/(1-(1+y)*x+y*(1-2*y)*x^2).
EXAMPLE
First five rows:
1;
0, 3;
0, 2, 5;
0, 2, 4, 11;
0, 2, 4, 14, 21;
First five polynomials u(n,x):
1
3x
2x + 5x^2
2x + 4x^2 + 11x^3
2x + 4x^2 + 14x^3 + 21x^4.
MATHEMATICA
u[1, x_] := 1; v[1, x_] := 1; z = 13;
u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];
v[n_, x_] := 2 x*u[n - 1, x] + x*v[n - 1, x];
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[%] (* A208328 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%] (* A208329 *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Feb 26 2012
STATUS
approved