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%I #20 Dec 28 2021 11:17:26
%S 1,1,1,1,3,1,1,7,3,1,1,15,6,5,1,1,31,10,16,5,1,1,63,15,42,15,7,1,1,
%T 127,21,99,35,29,7,1,1,255,28,219,70,93,28,9,1,1,511,36,466,126,256,
%U 84,46,9,1,1,1023,45,968,210,638,210,176,45,11,1,1,2047,55,1981,330,1486,462,562,165,67,11,1
%N Triangle read by rows, A007318 * A177990.
%C Row sums = A045623: (1, 2, 5, 12, 28, 64, 144, ...).
%C Double Riordan array ( 1/(1 - x); x/(1 - 2*x), x*(1 - 2*x)/(1 - x)^2 ) as defined in Davenport et al. - _Peter Bala_, Aug 25 2021
%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 As infinite lower triangular matrices, A007318 * A177990.
%F From _Peter Bala_, Aug 25 2021: (Start)
%F T(n,2*k) = T(n-1,2*k-1) - T(n-1,2*k+1).
%F T(n,2*k+1) = 2*T(n-1,2*k+1) + T(n-1,2*k).
%F G.f.: A(x,t) = (1 - t)/(1 - 2*t)*(1 - 2*t + t*x)/((1 - t)^2 - t^2*x^2) = 1 + (1 + x)*t + (1 + 3*x + x^2)^t^2 + ....
%F G.f. column 2*k: x^(2*k)/(1 - x)^(2*k+1).
%F G.f. column 2*k+1: x^(2*k+1)/((1 - x)^(2*k+1) * (1 - 2*x)). (End)
%e First few rows of the triangle:
%e 1;
%e 1, 1;
%e 1, 3, 1;
%e 1, 7, 3, 1;
%e 1, 15, 6, 5, 1;
%e 1, 31, 10, 16, 5, 1;
%e 1, 63, 15, 42, 15, 7, 1;
%e 1, 127, 21, 99, 35, 29, 7, 1;
%e 1, 255, 28, 219, 70, 93, 28, 9, 1;
%e 1, 511, 36, 466, 126, 256, 84, 46, 9, 1;
%e 1, 1023, 45, 968, 210, 638, 210, 176, 45, 11, 1;
%e 1, 2047, 55, 1981, 330, 1486, 462, 562, 165, 67, 11, 1;
%e 1, 4095, 66, 4017, 495, 3302, 924, 1586, 495, 299, 66, 13, 1;
%e ...
%Y Cf. A177993 = A177990 * A007318.
%Y Cf. A045623, A070909.
%K nonn,tabl
%O 0,5
%A _Gary W. Adamson_, May 16 2010
%E a(8) corrected and more terms by _Georg Fischer_, Dec 28 2021
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