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**The Hankel Transform and Some of its Properties**

### John W. Layman

Department of Mathematics

Virginia Polytechnic Institute and State University

Blacksburg, Virginia 24061, USA

Email address: layman@math.vt.edu

**Abstract:**
The Hankel transform of an integer sequence is defined
and some of its properties discussed.
It is shown that the Hankel transform of a sequence S is the
same as the Hankel transform of the
binomial or invert transform of S.
If H is the Hankel matrix of a sequence and H = LU is the LU
decomposition of H,
the behavior of the first super-diagonal of U under the binomial
or invert transform is also studied.
This leads to a simple classification scheme for certain integer sequences.

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(Concerned with sequences
A000079,
A000085,
A000108,
A000110,
A000142,
A000166,
A000178,
A000296,
A000522,
A000957,
A000984,
A001006,
A001405,
A001700,
A002212,
A002426,
A003701,
A005043,
A005425,
A005493,
A005494,
A005572,
A005773,
A007317,
A010483,
A010842,
A026375,
A026378,
A026569,
A026585,
A026671,
A033321,
A033543,
A045379,
A049027,
A052186,
A053486,
A053487,
A054341,
A054391,
A054393,
A055209,
A055878,
A055879,
A059738.)

Received May 3, 2001. Published in Journal of Integer Sequences, June 8, 2001.

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