OFFSET
1,2
COMMENTS
Alternating row sums: 1, 0, 0, 0, 0, 0, 0, 0, 0, ...
For a discussion and guide to related arrays, see A208510.
As triangle T(n,k) with 0 <= k <= n, it is (2, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (2, 0, 0, 0, 0, 0, 0, 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) = u(n-1,x) + 2x*v(n-1,x) + 1,
where u(1,x)=1, v(1,x)=1.
As triangle T(n,k) with 0 <= k <= n: T(n,k) = A029653(n,k)*2^k. - Philippe Deléham, Mar 04 2012
Sum_{k=0..n} T(n,k)*x^k = 2*(1+x)*(1+2x)^(n-2) for n > 1. - Philippe Deléham, Mar 05 2012
EXAMPLE
First five rows:
1;
2, 2;
2, 6, 4;
2, 10, 16, 8;
2, 14, 36, 40, 16;
First five polynomials v(n,x):
1
2 + 2x = 2*(1+x)
2 + 6x + 4x^2 = 2*(1+x)*(1+2x)
2 + 10x + 16x^2 + 8x^3 = 2*(1+x)*(1+2x)^2
2 + 14x + 36x^2 + 40x^3 + 16x^4 = 2*(1+x)*(1+2x)^3
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_] := u[n - 1, x] + 2 x*v[n - 1, x] + 1;
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[%] (* A185045 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%] (* A208659 *)
(* Using the function RiordanSquare defined in A321620 we also have: *)
A208659 = RiordanSquare[(1 + x)/(1 - x), 16] // Flatten (* Gerry Martens, Oct 16 2022 *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Mar 03 2012
STATUS
approved