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A177992 Triangle read by rows, A007318 * A177990.

%I A177992 #20 Dec 28 2021 11:17:26
%S A177992 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 A177992 127,21,99,35,29,7,1,1,255,28,219,70,93,28,9,1,1,511,36,466,126,256,
%U A177992 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 A177992 Triangle read by rows, A007318 * A177990.
%C A177992 Row sums = A045623: (1, 2, 5, 12, 28, 64, 144, ...).
%C A177992 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 A177992 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 A177992 As infinite lower triangular matrices, A007318 * A177990.
%F A177992 From _Peter Bala_, Aug 25 2021: (Start)
%F A177992 T(n,2*k) = T(n-1,2*k-1) - T(n-1,2*k+1).
%F A177992 T(n,2*k+1) = 2*T(n-1,2*k+1) + T(n-1,2*k).
%F A177992 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 A177992 G.f. column 2*k: x^(2*k)/(1 - x)^(2*k+1).
%F A177992 G.f. column 2*k+1: x^(2*k+1)/((1 - x)^(2*k+1) * (1 - 2*x)). (End)
%e A177992 First few rows of the triangle:
%e A177992   1;
%e A177992   1,    1;
%e A177992   1,    3,  1;
%e A177992   1,    7,  3,    1;
%e A177992   1,   15,  6,    5,   1;
%e A177992   1,   31, 10,   16,   5,    1;
%e A177992   1,   63, 15,   42,  15,    7,   1;
%e A177992   1,  127, 21,   99,  35,   29,   7,    1;
%e A177992   1,  255, 28,  219,  70,   93,  28,    9,   1;
%e A177992   1,  511, 36,  466, 126,  256,  84,   46,   9,   1;
%e A177992   1, 1023, 45,  968, 210,  638, 210,  176,  45,  11,  1;
%e A177992   1, 2047, 55, 1981, 330, 1486, 462,  562, 165,  67, 11,  1;
%e A177992   1, 4095, 66, 4017, 495, 3302, 924, 1586, 495, 299, 66, 13, 1;
%e A177992   ...
%Y A177992 Cf. A177993 = A177990 * A007318.
%Y A177992 Cf. A045623, A070909.
%K A177992 nonn,tabl
%O A177992 0,5
%A A177992 _Gary W. Adamson_, May 16 2010
%E A177992 a(8) corrected and more terms by _Georg Fischer_, Dec 28 2021