 Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.1

## The k-Binomial Transforms and the Hankel Transform

### Michael Z. Spivey Department of Mathematics and Computer Science University of Puget Sound Tacoma, Washington 98416-1043 USA Laura L. Steil Department of Mathematics and Computer Science Samford University Birmingham, Alabama 35229 USA

Abstract: We give a new proof of the invariance of the Hankel transform under the binomial transform of a sequence. Our method of proof leads to three variations of the binomial transform; we call these the \$k\$-binomial transforms. We give a simple means of constructing these transforms via a triangle of numbers. We show how the exponential generating function of a sequence changes after our transforms are applied, and we use this to prove that several sequences in the On-Line Encyclopedia of Integer Sequences are related via our transforms. In the process, we prove three conjectures in the OEIS. Addressing a question of Layman, we then show that the Hankel transform of a sequence is invariant under one of our transforms, and we show how the Hankel transform changes after the other two transforms are applied. Finally, we use these results to determine the Hankel transforms of several integer sequences.

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(Concerned with sequences A000012 A000032 A000045 A000079 A000108 A000142 A000165 A000166 A000354 A000364 A000609 A000984 A001003 A001006 A001653 A001907 A002078 A002315 A002426 A002801 A003645 A005799 A007052 A007070 A007680 A007696 A010844 A010845 A014445 A014448 A032031 A047053 A052562 A055209 A056545 A056546 A059231 A059304 A075271 A075272 A082032 A084770 A084771 A097814 A097815 and A097816 .)

Received June 24 2005; revised version received November 1 2005. Published in Journal of Integer Sequences November 15 2005.