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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, 127, 21, 99, 35, 29, 7, 1, 1, 255, 28, 219, 70, 93, 28, 9, 1, 1, 511, 36, 466, 126, 256, 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
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OFFSET
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0,5
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COMMENTS
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Row sums = A045623: (1, 2, 5, 12, 28, 64, 144, ...).
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
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LINKS
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D. E. Davenport, L. W. Shapiro and L. C. Woodson, The Double Riordan Group, The Electronic Journal of Combinatorics, 18(2) (2012).
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FORMULA
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T(n,2*k) = T(n-1,2*k-1) - T(n-1,2*k+1).
T(n,2*k+1) = 2*T(n-1,2*k+1) + T(n-1,2*k).
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 + ....
G.f. column 2*k: x^(2*k)/(1 - x)^(2*k+1).
G.f. column 2*k+1: x^(2*k+1)/((1 - x)^(2*k+1) * (1 - 2*x)). (End)
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EXAMPLE
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First few rows of the triangle:
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, 127, 21, 99, 35, 29, 7, 1;
1, 255, 28, 219, 70, 93, 28, 9, 1;
1, 511, 36, 466, 126, 256, 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;
1, 4095, 66, 4017, 495, 3302, 924, 1586, 495, 299, 66, 13, 1;
...
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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