OFFSET
1,2
COMMENTS
Alternating row sums: 1,2,2,2,2,2,2,2,2,2,2,2,2,...
For a discussion and guide to related arrays, see A208510.
As triangle T(n,k) with 0 <= k <= n, it is (3, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 08 2012
A skew triangle of A209144. - Philippe Deléham, Mar 08 2012
Riordan array ( (1 + x)/(1 - 2*x), x/(1 - 2*x) ). Cf. A118800. Matrix inverse is a signed version of A112626. - Peter Bala, Jul 17 2013
FORMULA
u(n,x) = u(n-1,x) + (x+1)*v(n-1,x),
v(n,x) = u(n-1,x) + (x+1)*v(n-1,x) + 1,
where u(1,x)=1, v(1,x)=1.
As DELTA-triangle:
T(n,k) = 2*T(n-1,k) + T(n-1,k-1), T(0,0) = 1, T(1,0) = 3, T(1,1) = 1, T(n,k) = 0 if k < 0 or if k > n. - Philippe Deléham, Mar 08 2012
As DELTA-triangle: G.f. is (1+x)/(1-2*x-yx). - Philippe Deléham, Mar 08 2012
EXAMPLE
First five rows:
1;
3, 1;
6, 5, 1;
12, 16, 7, 1;
24, 44, 30, 9, 1;
First three polynomials v(n,x): 1, 3 + x, 6 + 5x + x^2.
v(1,x) = 1
v(2,x) = 3 + x
v(3,x) = (3 + x)*(2 + x)
v(4,x) = (3 + x)*(2 + x)^2
v(5,x) = (3 + x)*(2 + x)^3
v(n,x) = (3 + x)*(2 + x)^(n-2)for n > 1. - Philippe Deléham, Mar 08 2012
MATHEMATICA
u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + (x + 1)*v[n - 1, x];
v[n_, x_] := u[n - 1, x] + (x + 1)*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[%] (* A209148 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%] (* A209149 *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Mar 07 2012
STATUS
approved