OFFSET
1,3
COMMENTS
Alternating row sums: 1,-1,-1,-1,-1,-1,-1,-1,...
For a discussion and guide to related arrays, see A208510.
Subtriangle of the triangle given by [1,0,1,0,0,0,0,...] DELTA [0,2,0,-1,0,0,0,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 04 2012
FORMULA
u(n,x) = u(n-1,x) + 2x*v(n-1,x),
v(n,x) = (x+1)*u(n-1,x) + (x+1)*v(n-1,x),
where u(1,x)=1, v(1,x)=1.
Recurrence: T(n,k) = 2*T(n-1,k) + T(n-1,k-1) - T(n-2,k) + T(n-2,k-1) + 2*T(n-2,k-2). - Philippe Deléham, Mar 04 2012
G.f.: (-1-x*y+x)*x*y/(-1+x*y+2*x+2*x^2*y^2+x^2*y-x^2). - R. J. Mathar, Aug 12 2015
EXAMPLE
First five rows:
1;
1, 2;
1, 6, 4;
1, 12, 18, 8;
1, 20, 52, 50, 16;
First five polynomials u(n,x):
1
1 + 2x
1 + 6x + 4x^2
1 + 12x + 18x^2 + 8x^3
1 + 20x + 52x^2 + 50x^3 + 16x^4
From Philippe Deléham, Mar 04 2012: (Start)
Triangle (1, 0, 1, 0, 0, 0, ...) DELTA (0, 2, 0, -1, 0, 0, ...) begins:
1;
1, 0;
1, 2, 0;
1, 6, 4, 0;
1, 12, 18, 8, 0;
1, 20, 52, 50, 16, 0;
1, 30, 120, 186, 126, 32, 0; (End)
MATHEMATICA
u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + 2 x*v[n - 1, x];
v[n_, x_] := (x + 1)*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[%] (* A208761 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%] (* A208762 *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Mar 03 2012
STATUS
approved