OFFSET
1,2
COMMENTS
row sums, u(n,1): A000129
row sums, v(n,1): A001333
alternating row sums, u(n,-1): 1,0,-1,-2,-3,-4,-5,-6,...
alternating row sums, v(n,-1): 1,1,1,1,1,1,1,1,1,1,1,...
Subtriangle of the triangle T(n,k) given by (1, 1, -1, 1, 0, 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, Mar 26 2012
FORMULA
u(n,x) = u(n-1,x) + x*v(n-1,x),
v(n,x) = (x+1)*u(n-1,x) + v(n-1,x),
where u(1,x)=1, v(1,x)=1.
From Philippe Deléham, Mar 26 2012: (Start)
As DELTA-triangle T(n,k) with 0 <= k <= n:
G.f.: (1-x+x^2-y^2*x^2)/(1-2*x+x^2-y*x^2-y^2*x^2).
T(n,k) = 2*T(n-1,k) - T(n-2,k) + T(n-2,k-1) + T(n-2,k-2), T(0,0) = T(1,0) = T(2,1) = 1, T(2,0) = 2, T(1,1) = T(2,2) = 0. (End)
EXAMPLE
First five rows:
1;
2, 1;
3, 3, 1;
4, 7, 5, 1;
5, 14, 15, 6, 1;
First five polynomials v(n,x):
1
2 + x
3 + 3x + x^2
4 + 7x + 5x^2 + x^3
5 + 14x + 15x^2 + 6x^3 + x^4
From Philippe Deléham, Mar 26 2012: (Start)
(1, 1, -1, 1, 0, 0, 0, ...) DELTA (0, 1, 0, -1, 0, 0, ...) begins:
1;
1, 0;
2, 1, 0;
3, 3, 1, 0;
4, 7, 5, 1, 0;
5, 14, 15, 6, 1, 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] + 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[%] (* A208334 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%] (* A208335 *)
Table[u[n, x] /. x -> 1, {n, 1, z}] (* u row sums *)
Table[v[n, x] /. x -> 1, {n, 1, z}] (* v row sums *)
Table[u[n, x] /. x -> -1, {n, 1, z}](* u alt. row sums *)
Table[v[n, x] /. x -> -1, {n, 1, z}](* v alt. row sums *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Feb 26 2012
STATUS
approved