OFFSET
1,5
COMMENTS
Subtriangle of the triangle given by (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 2, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Apr 09 2012
FORMULA
u(n,x) = u(n-1,x) + x*v(n-1,x),
v(n,x) = (x+1)*u(n-1,x) + 2x*v(n-1,x),
where u(1,x)=1, v(1,x)=1.
From Philippe Deléham, Apr 09 2012: (Start)
As DELTA-triangle T(n,k) with 0 <= k <= n:
G.f.: (1-2*y*x-y^2*x^2)/(1-x-2*y*x+y*x^2-y^2*x^2).
T(n,k) = T(n-1,k) + 2*T(n-1,k-1) - T(n-2,k-1) + T(n-2,k-2), T(0,0) = T(1,0) = T(2,0) = T(2,1) = 1, T(1,1) = T(2,2) = 0 and T(n,k) = 0 if k < 0 or if k > n. (End)
EXAMPLE
First five rows:
1;
1, 1;
1, 2, 3;
1, 3, 7, 7;
1, 4, 12, 20, 17;
First five polynomials u(n,x):
1
1 + x
1 + 2x + 3x^2
1 + 3x + 7x^2 + 7x^3
1 + 4x + 12x^2 + 20x^3 + 17x^4
From Philippe Deléham, Apr 09 2012: (Start)
(1, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 2, -1, 0, 0, 0, ...) begins:
1;
1, 0;
1, 1, 0;
1, 2, 3, 0;
1, 3, 7, 7, 0;
1, 4, 12, 20, 17, 0;
1, 5, 18, 40, 57, 41, 0; (End)
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 + 1)*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[%] (* A208338 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%] (* A208339 *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Feb 27 2012
STATUS
approved