Some canonical sequences of integers

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(ex − 1) + 1.

[1]  John Riordan,et al.  Introduction to Combinatorial Analysis , 1958 .

[2]  J. C. P. Miller,et al.  A method for the determination of converging factors, applied to the asymptotic expansions for the parabolic cylinder functions , 1952, Mathematical Proceedings of the Cambridge Philosophical Society.

[3]  Robert W. Robinson,et al.  LINE COLORED TREES WITH EXTENDABLE AUTOMORPHISMS , 1982 .

[4]  E. Netto,et al.  Lehrbuch der Combinatorik , 1902 .

[5]  Edward A. Bender,et al.  THE ENUMERATIVE USES OF GENERATING FUNCTIONS. , 1969 .

[6]  D. G. Rogers.  A schröder triangle: Three combinatorial problems , 1977 .

[7]  J. Touchard,et al.  Nombres Exponentiels Et Nombres De Bernoulli , 1956, Canadian Journal of Mathematics.

[8]  H. O. Foulkes.  Group transitivity and a multiplicative function of a partition , 1970 .

[9]  R. Guy,et al.  The Book of Numbers , 2019, The Crimean Karaim Bible.

[10]  M. Schader.  Hierarchical analysis: Classification with ordinal object dissimilarities , 1980 .

[11]  M. Waterman.  Secondary Structure of Single-Stranded Nucleic Acidst , 1978 .

[12]  G. Rota.  The Number of Partitions of a Set , 1964 .

[13]  G. Hardy,et al.  An Introduction to the Theory of Numbers , 1938 .

[14]  F D Murnaghan.  Airey's Converging Factor. , 1972, Proceedings of the National Academy of Sciences of the United States of America.

[15]  Irene A. Stegun,et al.  Handbook of Mathematical Functions. , 1966 .

[16]  Donald E. Knuth,et al.  Computation of Tangent, Euler, and Bernoulli Numbers* , 1967 .

[17]  H. Moses.  A kernel of Gel’fand–Levitan type for the three‐dimensional Schrödinger equation , 1980 .

[18]  Fred S. Roberts,et al.  Applied Combinatorics , 1984 .

[19]  Louis W. Shapiro,et al.  How to Guess a Generating Function , 1992, SIAM J. Discret. Math..

[20]  D. Knuth,et al.  Mathematics for the Analysis of Algorithms , 1999 .

[21]  N. Sloane.  A Handbook Of Integer Sequences , 1973 .

[22]  Claude E. Shannon,et al.  The Number of Two‐Terminal Series‐Parallel Networks , 1942 .

[23]  Tamás Lengyel,et al.  On a Recurrence involving Stirling Numbers , 1984, Eur. J. Comb..

[24]  L. Carlitz.  Solution of Certain Recurrences , 1969 .

[25]  Robert Donaghey.  Binomial Self-Inverse Sequences and Tangent Coefficients , 1976, J. Comb. Theory, Ser. A.

[26]  Robert Donaghey,et al.  Motzkin Numbers , 1977, J. Comb. Theory, Ser. A.

[27]  G. Rota,et al.  Finite operator calculus , 1975 .

[28]  Aviezri S. Fraenkel,et al.  Cayley permutations , 1984, Discret. Math..

[29]  L. Carlitz,et al.  The number of labeled two-terminal series-parallel networks , 1956 .

[30]  都筑 俊郎.  On multiple transitivity of permutation groups , 1961 .

[31]  G. Kreweras.  Sur les hiérarchies de segments , 1973 .

[32]  John Riordan,et al.  Introduction to Combinatorial Analysis , 1959 .

[33]  John R. Airey D.Sc. Sc.D..  LII. The “converging factor” in asymptotic series and the calculation of Bessel, laguerre and other functions , 1937 .

[34]  I. M. H. Etherington.  XV.—On Non-Associative Combinations , 1940 .

[35]  Theodore S. Motzkin,et al.  SORTING NUMBERS FOR CYLINDERS AND OTHER CLASSICATION NUMBER , 1971 .

[36]  Peter J. Cameron,et al.  SOME TREELIKE OBJECTS , 1987 .

[37]  M. Kendall,et al.  Symmetric Function and Allied Tables. , 1967 .

[38]  Arthur Cayley,et al.  The Collected Mathematical Papers: On the analytical forms called trees , 1881 .

[39]  Robert Donaghey.  Automorphisms on Catalan trees and bracketings , 1980, J. Comb. Theory, Ser. B.

[40]  Adalbert Kerber.  A matrix of combinatorial numbers related to the symmetric groups , 1978, Discret. Math..

[41]  Michael S. Waterman,et al.  COMPUTATION OF GENERATING FUNCTIONS FOR BIOLOGICAL MOLECULES , 1980 .

[42]  Robert Donaghey,et al.  Restricted plane tree representations of four Motzkin-Catalan equations , 1977, J. Comb. Theory, Ser. B.

[43]  G. Pólya,et al.  Problems and theorems in analysis , 1983 .

[44]  Neil J. A. Sloane,et al.  The encyclopedia of integer sequences , 1995 .

[45]  Michael S. Waterman,et al.  On some new sequences generalizing the Catalan and Motzkin numbers , 1979, Discret. Math..

[46]  Peter J. Cameron,et al.  Some sequences of integers , 1989, Discret. Math..

[47]  Donald E. Knuth.  Convolution polynomials , 1992 .

[48]  Frank Harary,et al.  Twenty-step algorithm for determining the asymptotic number of trees of various species: corrigenda , 1986, Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics.

[49]  T. Motzkin,et al.  Relations between hypersurface cross ratios, and a combinatorial formula for partitions of a polygon, for permanent preponderance, and for non-associative products , 1948 .

[50]  P. Wynn.  Converging factors for the weber parabolic cylinder functions of complex argument, Ia , 1963 .

[51]  Z. A. Lomnicki.  Two-terminal series-parallel networks , 1972, Advances in Applied Probability.

[52]  Frank Harary,et al.  On the cell-growth problem for arbitrary polygons , 1975, Discret. Math..

[53]  John Riordan,et al.  Inverse Relations and Combinatorial Identities , 1964 .