Some Canonical Sequences of Integers

M. Bernstein(*) and N. J. A. Sloane(**) 

Mathematical Sciences Research Center 
AT&T Bell Laboratories 
Murray Hill, NJ 07974 USA 

(*)Present address:
Mathematics Department
Univ. California Berkeley
887 Evans Hall
Berkeley CA 94720-3840
Email: mira@math.berkeley.edu

(**)Present address:
Information Sciences Research
AT&T Shannon Lab
Florham Park, NJ 07932-0971 USA
Email: njas@research.att.com

Dedicated to Professor J. J. Seidel


Abstract

Extending earlier work of R. Donaghey and P. J. Cameron, we investigate
some canonical "eigen-sequences" associated with transformations of integer sequences.
Several known sequences appear in a new setting:
for instance the sequences (such as 1, 3, 11, 49, 257, 1531, ...) studied by
T. Tsuzuku, H. O. Foulkes and A. Kerber in connection with
multiply transitive groups are eigen-sequences for the binomial transform.
Many interesting new sequences also arise, such as 1, 1, 2, 26, 152, 1144, ...,
which shifts one place left when transformed by the Stirling numbers of the second kind,
and whose exponential generating function satisfies A'(x) = A(e^x -1) + 1.


For the full version see
http://www.research.att.com/~njas/doc/eigen.pdf (pdf) or
http://www.research.att.com/~njas/doc/eigen.ps (ps)


This paper was published (in a somewhat different form) in
Linear Algebra and Its Applications, Vol. 226/228 (The Seidel Festschrift) (1995),
pp. 57-72. [Math. Rev. 96i:05004]. Erratum: Linear Algebra Appl. 320 (2000), no. 1-3, 210.