OFFSET
1,4
COMMENTS
Eigensequence of the triangle = A158943: (1, 1, 3, 5, 10, 19, 36, 69, 131, ...)
LINKS
D. E. Davenport, L. W. Shapiro and L. C. Woodson, The Double Riordan Group, The Electronic Journal of Combinatorics, 18(2) (2012).
FORMULA
Triangle by columns: A027656: (1, 0, 2, 0, 3, 0, 4, 0, 5, ...) in every column.
From Peter Bala, Aug 15 2021: (Start)
T(n,k) = (1/2)*(n - k + 2) * (1 + (-1)^(n-k))/2 for 0 <= k <= n.
Double Riordan array (1/(1-x)^2; x, x) as defined in Davenport et al.
The m-th power of the array is the double Riordan array (1/(1 - x)^(2*m); x, x). Cf. A156663. (End)
EXAMPLE
First few rows of the triangle =
1;
0, 1;
2, 0, 1;
0, 2, 0, 1;
3, 0, 2, 0, 1;
0, 3, 0, 2, 0, 1;
4, 0, 3, 0, 2, 0, 1;
0, 4, 0, 3, 0, 2, 0, 1;
5, 0, 4, 0, 3, 0, 2, 0, 1;
0, 5, 0, 4, 0, 3, 0, 2, 0, 1;
6, 0, 5, 0, 4, 0, 3, 0, 2, 0, 1;
0, 6, 0, 5, 0, 4, 0, 3, 0, 2, 0, 1;
7, 0, 6, 0, 5, 0, 4, 0, 3, 0, 2, 0, 1;
...
The inverse array begins
1;
0, 1;
-2, 0, 1;
0, -2, 0, 1;
1, 0, -2, 0, 1;
0, 1, 0, -2, 0, 1;
0, 0, 1, 0, -2, 0, 1;
0, 0, 0, 1, 0, -2, 0, 1;
0, 0, 0, 0, 1, 0, -2, 0, 1;
... - Peter Bala, Aug 15 2021
MAPLE
seq(seq((1/2)*(n - k + 2) * (1 + (-1)^(n-k))/2, k = 0..n), n = 0..10) # Peter Bala, Aug 15 2021
CROSSREFS
KEYWORD
AUTHOR
Gary W. Adamson, Mar 31 2009
STATUS
approved