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COMMENTS
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See A177427 (numerators) for the description of the Akiyama-Tanigawa array of this sequence of fractions, T(0,k) = 1, 1, 13/12, 7/6, 149/120, 157/120, ...
If we add a zero in front and construct an array A(n,k) with successive differences A(n,k) = A(n-1,k+1)-A(n-1,k), the array A(.,.) becomes
0, 1, 1, 13/12, 7/6, 149/120, 157/120, 383/280, 199/140, 7409/5040, ...
1, 0, 1/12, 1/12, 3/40, 1/15, 5/84, 3/56, 7/144, 2/45, 9/220, ...
-1, 1/12, 0, -1/120, -1/120, -1/140, -1/168, -5/1008, -1/240, -7/1980, ...
13/12, -1/12, -1/120, 0, 1/840, 1/840, 1/1008, 1/1260, 1/1584, ...
-7/6, 3/40, 1/120, 1/840, 0, -1/5040, -1/5040, -1/6160, -1/7920, ...
149/120, -1/15, -1/140, -1/840, -1/5040, 0, 1/27720, 1/27720, ...
-157/120, 5/84, 1/168, 1/1008, 1/5040, 1/27720, 0, -1/144144, -1/144144, ...
On the diagonal, A(n,n)=0. The left column A(n,0) = (-1)^(n+1)*A(0,k) is a signed variant of the top row, which means the sequence is some eigensequence under the inverse binomial transform (see A174341 for other examples). This eigen-feature would remain if the same number of top rows and left columns were removed from A(.,.).
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