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%I #16 Aug 27 2021 16:34:00
%S 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,
%T 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,
%U 0,5,0,4,0,3,0,2,0,1
%N Triangle by columns: the natural numbers interleaved with zeros in every column: (1, 0, 2, 0, 3, 0, 4, ...)
%C Eigensequence of the triangle = A158943: (1, 1, 3, 5, 10, 19, 36, 69, 131, ...)
%H D. E. Davenport, L. W. Shapiro and L. C. Woodson, <a href="https://doi.org/10.37236/2034">The Double Riordan Group</a>, The Electronic Journal of Combinatorics, 18(2) (2012).
%F Triangle by columns: A027656: (1, 0, 2, 0, 3, 0, 4, 0, 5, ...) in every column.
%F From _Peter Bala_, Aug 15 2021: (Start)
%F T(n,k) = (1/2)*(n - k + 2) * (1 + (-1)^(n-k))/2 for 0 <= k <= n.
%F Double Riordan array (1/(1-x)^2; x, x) as defined in Davenport et al.
%F The m-th power of the array is the double Riordan array (1/(1 - x)^(2*m); x, x). Cf. A156663. (End)
%e First few rows of the triangle =
%e 1;
%e 0, 1;
%e 2, 0, 1;
%e 0, 2, 0, 1;
%e 3, 0, 2, 0, 1;
%e 0, 3, 0, 2, 0, 1;
%e 4, 0, 3, 0, 2, 0, 1;
%e 0, 4, 0, 3, 0, 2, 0, 1;
%e 5, 0, 4, 0, 3, 0, 2, 0, 1;
%e 0, 5, 0, 4, 0, 3, 0, 2, 0, 1;
%e 6, 0, 5, 0, 4, 0, 3, 0, 2, 0, 1;
%e 0, 6, 0, 5, 0, 4, 0, 3, 0, 2, 0, 1;
%e 7, 0, 6, 0, 5, 0, 4, 0, 3, 0, 2, 0, 1;
%e ...
%e The inverse array begins
%e 1;
%e 0, 1;
%e -2, 0, 1;
%e 0, -2, 0, 1;
%e 1, 0, -2, 0, 1;
%e 0, 1, 0, -2, 0, 1;
%e 0, 0, 1, 0, -2, 0, 1;
%e 0, 0, 0, 1, 0, -2, 0, 1;
%e 0, 0, 0, 0, 1, 0, -2, 0, 1;
%e ... - _Peter Bala_, Aug 15 2021
%p seq(seq((1/2)*(n - k + 2) * (1 + (-1)^(n-k))/2, k = 0..n), n = 0..10) # _Peter Bala_, Aug 15 2021
%Y Cf. A158943, A158945, A156663.
%K nonn,tabl,easy
%O 1,4
%A _Gary W. Adamson_, Mar 31 2009
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