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COMMENT
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Prefaced with a 1: (1, 1, 2, 5, 15, 51,...), convolved with A006789 =
A006789; identical to reversing k terms of one sequence then taking the dot product of
k terms of the other: e.g. A006789(6) = 143 = (51, 15, 5, 2, 1, 1) dot (1, 1, 2, 5, 14, 43)
= (51 + 15 + 10 + 10 + 14 + 43). Equals row sums of triangle A153199.
A153197 can be generated from the Hankel transform [1,1,1,...] by taking
successive iterates of the operations: (binomial transform of [1,1,1,...]
followed by INVERT transform of the result, then binomial transform of the
result, (repeat cycle)...; until the operations converge upon a two sequence
fixed limit cycle of A006789 and A153197. Or, the infinite set of operations Q may begin: INVERT transform of [1,1,1,..] followed by binomial transform of
the result, INVERT transform of the result, etc; until the operations again
converge upon A006789 and A153197. The two sequences A006789 and A153197 have the mutual relationships that
binomial transform of A006789 = A153197; while the INVERT transform of A153197 prefaced with a 1 = A006789.
Product of the two sequences (A153197 prefaced with a 1) and A006789
= A006789 with offset 1. Or, (1,1,2,5,15,51,...) * (1,1,2,5,14,43,...) = (1,2,5,14,43,...).
Conjecture: Given any sequence with Hankel transform of [1,1,1,...],
performing alternate operations: binomial transform followed by INVERT
transform, then binomial transform of the last result (repeat); or INVERT
transform starting first, will converge upon A006789 and A153197 as a two sequence limit cycle. The conjecture can be extended to any Hankel transform
(and their accompanying sequence set): analogous operations will converge
upon a Bessel-type sequence and its binomial transform.
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FORMULA
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Binomial transform of A006789, Bessel numbers (1, 1, 2, 5, 14, 43, 143, 509...) INVERTi transform of A006789, Bessel numbers with offset 1: (1, 2, 5, 14, 43, 143,...).
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