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COMMENTS
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The first 6 rows if the table generated by iterative application of the Akiyama-Tanigawa transform starting with a header row of fractions A164555(k)/A027642(k) are:
1, 1/2, 1/6, 0, -1/30, 0, 1/42, 0, -1/30, 0, 5/66, 0, -691/2730, 0, 7/6, ...
1/2, 2/3, 1/2, 2/15, -1/6, -1/7, 1/6, 4/15, -3/10, -25/33, 5/6, 1382/455, ...
-1/6, 1/3, 11/10, 6/5, -5/42, -13/7, -7/10, 68/15, 453/110, -175/11, ...
-1/2, -23/15, -3/10, 554/105, 365/42, -243/35, -1099/30, 548/165, 19827/110, ...
31/30, -37/15, -1171/70, -478/35, 469/6, 1247/7, -6153/22, -46708/33, ...
7/2, 599/21, -129/14, -38566/105, -20995/42, 211515/77, 524699/66, ...
The numerators of the leftmost column define the current sequence.
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