OFFSET
1,5
COMMENTS
For a discussion and guide to related arrays, see A208510.
Subtriangle of the triangle given by (1, 0, 2, -2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Apr 01 2012
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..1225
FORMULA
u(n,x) = x*u(n-1,x) + 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, Apr 01 2012: (Start)
As DELTA-triangle T(n,k) with 0 <= k <= n:
G.f.: (1-2*y*x-2*y*x^2+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, 4, 1;
1, 7, 9, 1;
1, 10, 26, 16, 1;
First three polynomials v(n,x):
1
1 + x
1 + 4x + x^2.
From Philippe Deléham, Apr 01 2012: (Start)
(1, 0, 2, -2, 0, 0, 0, ...) DELTA (0, 1, 0, 1, 0, 0, 0, ...) begins:
1;
1, 0;
1, 1, 0;
1, 4, 1, 0;
1, 7, 9, 1, 0;
1, 10, 26, 16, 1, 0;
1, 13, 52, 70, 25, 1, 0; (End)
MATHEMATICA
u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n - 1, x] + 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[%] (* A209414 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%] (* A112351 *)
CoefficientList[CoefficientList[Series[(1 - 2*y*x - 2*y*x^2 + y^2*x^2)/(1 - x - 2*y*x - y*x^2 + y^2*x^2), {x, 0, 10}, {y, 0, 10}], x], y] // Flatten (* G. C. Greubel, Jan 03 2018 *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Mar 09 2012
STATUS
approved