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A158944 Triangle by columns: the natural numbers interleaved with zeros in every column: (1, 0, 2, 0, 3, 0, 4, ...)

%I A158944 #16 Aug 27 2021 16:34:00
%S A158944 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 A158944 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 A158944 0,5,0,4,0,3,0,2,0,1
%N A158944 Triangle by columns: the natural numbers interleaved with zeros in every column: (1, 0, 2, 0, 3, 0, 4, ...)
%C A158944 Eigensequence of the triangle = A158943: (1, 1, 3, 5, 10, 19, 36, 69, 131, ...)
%H A158944 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 A158944 Triangle by columns: A027656: (1, 0, 2, 0, 3, 0, 4, 0, 5, ...) in every column.
%F A158944 From _Peter Bala_, Aug 15 2021: (Start)
%F A158944 T(n,k) = (1/2)*(n - k + 2) * (1 + (-1)^(n-k))/2 for 0 <= k <= n.
%F A158944 Double Riordan array (1/(1-x)^2; x, x) as defined in Davenport et al.
%F A158944 The m-th power of the array is the double Riordan array (1/(1 - x)^(2*m); x, x). Cf. A156663. (End)
%e A158944 First few rows of the triangle =
%e A158944   1;
%e A158944   0, 1;
%e A158944   2, 0, 1;
%e A158944   0, 2, 0, 1;
%e A158944   3, 0, 2, 0, 1;
%e A158944   0, 3, 0, 2, 0, 1;
%e A158944   4, 0, 3, 0, 2, 0, 1;
%e A158944   0, 4, 0, 3, 0, 2, 0, 1;
%e A158944   5, 0, 4, 0, 3, 0, 2, 0, 1;
%e A158944   0, 5, 0, 4, 0, 3, 0, 2, 0, 1;
%e A158944   6, 0, 5, 0, 4, 0, 3, 0, 2, 0, 1;
%e A158944   0, 6, 0, 5, 0, 4, 0, 3, 0, 2, 0, 1;
%e A158944   7, 0, 6, 0, 5, 0, 4, 0, 3, 0, 2, 0, 1;
%e A158944   ...
%e A158944 The inverse array begins
%e A158944    1;
%e A158944    0,  1;
%e A158944   -2,  0,  1;
%e A158944    0, -2,  0,  1;
%e A158944    1,  0, -2,  0,  1;
%e A158944    0,  1,  0, -2,  0,  1;
%e A158944    0,  0,  1,  0, -2,  0,  1;
%e A158944    0,  0,  0,  1,  0, -2,  0,  1;
%e A158944    0,  0,  0,  0,  1,  0, -2,  0,  1;
%e A158944    ... - _Peter Bala_, Aug 15 2021
%p A158944 seq(seq((1/2)*(n - k + 2) * (1 + (-1)^(n-k))/2, k = 0..n), n = 0..10) # _Peter Bala_, Aug 15 2021
%Y A158944 Cf. A158943, A158945, A156663.
%K A158944 nonn,tabl,easy
%O A158944 1,4
%A A158944 _Gary W. Adamson_, Mar 31 2009