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A156663 Triangle by columns, powers of 2 interleaved with zeros.

%I A156663 #17 Aug 27 2021 16:40:16
%S A156663 1,0,1,2,0,1,0,2,0,1,4,0,2,0,1,0,4,0,2,0,1,8,0,4,0,2,0,1,0,8,0,4,0,2,
%T A156663 0,1,16,0,8,0,4,0,2,0,1,0,16,0,8,0,4,0,2,0,1,32,0,16,0,8,0,4,0,2,0,1,
%U A156663 0,32,0,16,0,8,0,4,0,2,0,1
%N A156663 Triangle by columns, powers of 2 interleaved with zeros.
%C A156663 Eigensequence of the triangle = A001045.
%H A156663 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 A156663 Triangle by columns, (1, 0, 2, 0, 4, 0, 8, ...) in every column.
%F A156663 From _Peter Bala_, Aug 15 2021: (Start)
%F A156663 T(n,k) = sqrt(2)^((n - k)/2) * (1 + (-1)^(n-k))/2 for 0 <= k <= n.
%F A156663 Double Riordan array (1/(1 - 2*x^2); x, x) as defined in Davenport et al.
%F A156663 The m-th power of the array is the double Riordan array (1/(1 - 2*x^2)^(m); x, x). Cf. A158944. (End)
%e A156663 First few rows of the triangle =
%e A156663    1;
%e A156663    0,  1;
%e A156663    2,  0,  1;
%e A156663    0,  2,  0,  1;
%e A156663    4,  0,  2,  0,  1;
%e A156663    0,  4,  0,  2,  0,  1;
%e A156663    8,  0,  4,  0,  2,  0,  1;
%e A156663    0,  8,  0,  4,  0,  2,  0,  1;
%e A156663   16,  0,  8,  0,  4,  0,  2,  0,  1;
%e A156663    0, 16,  0,  8,  0,  4,  0,  2,  0,  1;
%e A156663   32,  0, 16,  0,  8,  0,  4,  0,  2,  0,  1;
%e A156663    0, 32,  0, 16,  0,  8,  0,  4,  0,  2,  0,  1;
%e A156663   ...
%e A156663 The inverse array begins
%e A156663    1;
%e A156663    0,  1;
%e A156663   -2,  0,  1;
%e A156663    0, -2,  0,  1;
%e A156663    0,  0, -2,  0,  1;
%e A156663    0,  0,  0, -2,  0,  1;
%e A156663    0,  0,  0,  0, -2,  0,  1;
%e A156663    0,  0,  0,  0,  0, -2,  0,  1;
%e A156663    0,  0,  0,  0,  0,  0, -2,  0,  1;
%e A156663    ... - _Peter Bala_, Aug 15 2021
%p A156663 seq(seq( sqrt(2)^(n-k) * (1 + (-1)^(n-k))/2, k = 0..n), n = 0..10) # _Peter Bala_, Aug 15 2021
%Y A156663 Cf. A001045, A158944.
%K A156663 nonn,tabl
%O A156663 0,4
%A A156663 _Gary W. Adamson_, Feb 12 2009
%E A156663 Typo in Data corrected by _Peter Bala_, Aug 15 2021