OFFSET
1,3
COMMENTS
Alternating row sums: 1,-2,2,-2,2,-2,2,-2,...
For a discussion and guide to related arrays, see A208510.
Subtriangle of the triangle given by (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 3, -2/3, -1/3, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 24 2012
Mirror image of triangle in A054458. - Philippe Deléham, Mar 24 2012
FORMULA
u(n,x) = x*u(n-1,x) + (x+1)*v(n-1,x),
v(n,x) = 2x*u(n-1,x) + (x+1)*v(n-1,x),
where u(1,x)=1, v(1,x)=1.
From Philippe Deléham, Mar 24 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) = 1, T(1,1) = T(2,2) = 0, T(2,1) = 3 and T(n,k) = 0 if k < 0 or if k > n. (End)
EXAMPLE
First five rows:
1;
1, 3;
1, 6, 7;
1, 9, 23, 17;
1, 12, 48, 76, 41;
First three polynomials v(n,x):
1
1 + 3x
1 + 6x + 7x^2.
From Philippe Deléham, Mar 24 2012: (Start)
(1, 0, 0, 0, 0, ...) DELTA (0, 3, -2/3, -1/3, 0, 0, ...) begins:
1;
1, 0;
1, 3, 0;
1, 6, 7, 0;
1, 9, 23, 17, 0;
1, 12, 48, 76, 41, 0; (End)
MATHEMATICA
u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n - 1, x] + (x + 1)*v[n - 1, x];
v[n_, x_] := 2 x*u[n - 1, x] + (x + 1)*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[%] (* A209695 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%] (* A209696 *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Mar 13 2012
STATUS
approved