OFFSET
1,3
COMMENTS
row sums, u(n,1): (1,2,5,13,...), odd-indexed Fibonacci numbers
row sums, v(n,1): (1,3,8,21,...), even-indexed Fibonacci numbers
As triangle T(n,k) with 0<=k<=n, it is (0, 1/3, 2/3, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (3, -2/3, -1/3, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Feb 26 2012
FORMULA
u(n,x)=u(n-1,x)+x*v(n-1,x),
v(n,x)=x*u(n-1,x)+2x*v(n-1,x),
where u(1,x)=1, v(1,x)=1.
(Start)- As triangle T(n,k), 0<=k<=n :
T(n,k) = T(n-1,k) + 2*T(n-1,k-1) + T(n-2,k-2) - 2*T(n-2,k-1) with T(0,0) = 1, T(1,0) = 0, T(1,1) = 3, T(n,k) = 0 if k<0 or if k>n.
G.f.: (1+(y-1)*x)/(1-(1+2*y)*x+y*(2-y)*x^2).
EXAMPLE
First five rows:
1
0...3
0...1...7
0...1...3...17
0...1...3...10...41
First five polynomials u(n,x):
1, 3x, x + 7x^2, x + 3x^2 + 17x^3, x + 3x^2 + 10x^3 +
41x^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_] := x*u[n - 1, x] + 2 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[%] (* A208344 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%] (* A208345 *)
Table[u[n, x] /. x -> 1, {n, 1, z}]
Table[v[n, x] /. x -> 1, {n, 1, z}]
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Feb 25 2012
STATUS
approved