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The Encyclopedia of Integer Sequences
"... This article gives a brief introduction to the OnLine Encyclopedia of Integer Sequences (or OEIS). The OEIS is a database of nearly 90,000 sequences of integers, arranged lexicographically. The entry for a sequence lists the initial terms (50 to 100, if available), a description, formulae, programs ..."
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Cited by 631 (15 self)
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This article gives a brief introduction to the OnLine Encyclopedia of Integer Sequences (or OEIS). The OEIS is a database of nearly 90,000 sequences of integers, arranged lexicographically. The entry for a sequence lists the initial terms (50 to 100, if available), a description, formulae, programs to generate the sequence, references, links to relevant web pages, and other
Automatic Invention of Integer Sequences
, 2000
"... We report on the application of the HR program (Colton, Bundy, & Walsh 1999) to the problem of automatically inventing integer sequences. Seventeen sequences invented by HR are interesting enough to have been accepted into the Encyclopedia of Integer Sequences (Sloane 2000) and all were supplie ..."
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Cited by 28 (16 self)
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We report on the application of the HR program (Colton, Bundy, & Walsh 1999) to the problem of automatically inventing integer sequences. Seventeen sequences invented by HR are interesting enough to have been accepted into the Encyclopedia of Integer Sequences (Sloane 2000) and all were supplied with interesting conjectures about their nature, also discovered by HR. By extending HR, we have enabled it to perform a two stage process of invention and investigation. This involves generating both the definition and terms of a new sequence, relating it to sequences already in the Encyclopedia and pruning the output to help identify the most surprising and interesting results.
Oneparameters groups and combinatorial physics
 Proceedings of the Symposium COPROMAPH3: Contemporary Problems in Mathematical Physics (PortoNovo
, 2003
"... In this communication, we consider the normal ordering of operators of the type ..."
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Cited by 11 (9 self)
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In this communication, we consider the normal ordering of operators of the type
Hierarchical Dobińskitype relations via substitution and the moment problem, J.Phys
 A: Math.Gen
, 2004
"... Abstract. We consider the transformation properties of integer sequences arising from the normal ordering of exponentiated boson ([a, a † ] = 1) monomials of the form exp[λ(a † ) r a], r = 1, 2,..., under the composition of their exponential generating functions (egf). They turn out to be of Sheffe ..."
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Cited by 5 (3 self)
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Abstract. We consider the transformation properties of integer sequences arising from the normal ordering of exponentiated boson ([a, a † ] = 1) monomials of the form exp[λ(a † ) r a], r = 1, 2,..., under the composition of their exponential generating functions (egf). They turn out to be of Sheffertype. We demonstrate that two key properties of these sequences remain preserved under substitutional composition: a) the property of being the solution of the Stieltjes moment problem; and b) the representation of these sequences through infinite series (Dobińskitype relations). We present a number of examples of such composition satisfying properties a) and b). We obtain new Dobińskitype formulas and solve the associated moment problem for several hierarchically defined combinatorial families of sequences. Hierarchical Dobińskitype relations via substitution and the moment problem 2 1.
A combinatorial interpretation for the eigensequence for composition
 Journal of Integer Sequences
"... The monic sequence that shifts left under convolution with itself is the Catalan numbers with 130+ combinatorial interpretations. Here we establish a combinatorial interpretation for the monic sequence that shifts left under composition: it counts permutations that contain a 3241 pattern only as par ..."
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Cited by 4 (1 self)
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The monic sequence that shifts left under convolution with itself is the Catalan numbers with 130+ combinatorial interpretations. Here we establish a combinatorial interpretation for the monic sequence that shifts left under composition: it counts permutations that contain a 3241 pattern only as part of a 35241 pattern. We give two recurrences, the first allowing relatively fast computation, the second similar to one for the Catalan numbers. Among the 4 × 4! = 96 similarly restricted patterns involving 4 letters (such as 4231: a 431 pattern occurs only as part of a 4231), four different counting sequences arise: 64 give the Catalan numbers, 16 give the Bell numbers, 12 give sequence A051295 in OEIS, and 4 give a new sequence with an explicit formula. 1
Some Easily Derivable Integer Sequences
 J. Integer Seq
, 2000
"... We propose and discuss several simple ways of obtaining new enumerative sequences from existing ones. For instance, the number of graphs considered up to the action of an involutory transformation is expressible as the semisum of the total number of such graphs and the number of graphs invariant un ..."
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Cited by 3 (1 self)
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We propose and discuss several simple ways of obtaining new enumerative sequences from existing ones. For instance, the number of graphs considered up to the action of an involutory transformation is expressible as the semisum of the total number of such graphs and the number of graphs invariant under the involution. Another, less familiar idea concerns even and oddedged graphs: the di#erence between their numbers often proves to be a very simple quantity (such as n!). More than 30 new sequences will be constructed by these methods.
Stories About Groups and Sequences
"... The main theme of this article is that counting orbits of an infinite permutation group on finite subsets or tuples is very closely related to combinatorial enumeration; this point of view ties together various disparate "stories". 1 1 Twographs and even graphs The first story originated with ..."
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Cited by 1 (0 self)
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The main theme of this article is that counting orbits of an infinite permutation group on finite subsets or tuples is very closely related to combinatorial enumeration; this point of view ties together various disparate "stories". 1 1 Twographs and even graphs The first story originated with Neil Sloane, when he was compiling the first edition of his dictionary of integer sequences [35]. He observed that certain counting sequences appeared to agree. The first sequence enumerates even graphs, those in which any vertex has even valency (so that the graph is a disjoint union of Eulerian graphs). These graphs were enumerated by Robinson [29] and Liskovec [18]. The second sequence counts switching classes of graphs. If \Gamma is a graph on the vertex set X, and Y is a subset of X, the result of switching \Gamma with respect to Y is obtained by deleting all edges between Y and its complement, putting in all edges between Y and its complement which didn't exist before, and leaving t...
Automatic Invention of Integer Sequences
, 2000
"... We report on the application of the HR program (Colton, Bundy, & Walsh 1999) to the problem of automatically inventing integer sequences. Seventeen sequences invented by HR are interesting enough to have been accepted into the Encyclopedia of Integer Sequences (Sloane 2000) and all were supplie ..."
Abstract
 Add to MetaCart
We report on the application of the HR program (Colton, Bundy, & Walsh 1999) to the problem of automatically inventing integer sequences. Seventeen sequences invented by HR are interesting enough to have been accepted into the Encyclopedia of Integer Sequences (Sloane 2000) and all were supplied with interesting conjectures about their nature, also discovered by HR. By extending HR, we have enabled it to perform a two stage process of invention and investigation. This involves generating both the definition and terms of a new sequence, relating it to sequences already in the Encyclopedia and pruning the output to help identify the most surprising and interesting results.