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Applied mathematics (18)

Discrete mathematics (18)Proceedings (1)Pure mathematics (1)View MoreJournalBulletin of the Institute of Combinatorics and its Applications (ICA) (7)Fibonacci Quarterly (6)Utilitas Mathematica (3)Carpathian Journal of Mathematics (1)Electronic Notes in Discrete Mathematics (1)AuthorsBagdasar, Ovidiu (2)01f9023f-d29d-4a5e-9798-89f4c5851ab9 (1)81a26182-8169-4390-b85f-b5fad16813cb (1)a90cc4fe-8caf-4587-ab1d-ca6360cecf5b (1)afaa2e14-da18-425a-9673-04ed98ca29b9 (1)View MoreYear (Issue Date)2014-05 (4)2016-08 (2)2014 (1)2014-02 (1)2015 (1)View MoreTypesArticle (18)
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The Asymptotic Form of the Sum $\sum_{i=0}^{n} i^{p} { n+i \choose i }$: Two Proofs

Larcombe, Peter J.; Kirschenhofer, Peter; Fennessey, Eric J. (The Institute of Combinatorics and its Applications, 2014)

Generalised Catalan polynomials and their properties

Larcombe, Peter J.; Jarvis, Frazer A.; Fennessey, Eric J. (The Institute of Combinatorics and its Applications, 2014-05)

We introduce a new type of polynomial, termed a generalised Catalan polynomial. We list essential mathematical properties and give two associated combinatorial interpretations.

Closed form evaluations of some series involving Catalan numbers

Larcombe, Peter J. (The Institute of Combinatorics and its Applications (ICA), 2014-05)

Closed form evaluations of some infinite series comprising sums of exponentiated multiples of Catalan numbers are yielded by a known expansion of the trigonometric function sin(2α), and then re-formulated independently as verification.

On cyclicity and density of some Catalan polynomial sequences

Larcombe, Peter J.; Fennessey, Eric J. (The Institute of Combinatorics and its Applications, 2014-05)

We give proofs of cyclic and density properties of some sequences generated by Catalan polynomials, and other observations.

A polynomial based construction of periodic Horadam sequences

Larcombe, Peter J.; Fennessey, Eric J. (The Institute of Combinatorics and its Applications (ICA), 2016-03)

A recent matrix based approach to the study of self-repeating Horadam sequences has identified a mechanism to produce guaranteed (and arbitrary) periodicity through a novel formulation for the two governing parameters in the defining linear Horadam recurrence equation. We consider this further here, giving supporting examples to illustrate the methodology which utilises so called Catalan polynomials.

Some factorisation and divisibility properties of Catalan polynomials

Jarvis, Frazer A.; Larcombe, Peter J.; Fennessey, Eric J. (The Institute of Combinatorics and its Applications, 2014-05)

We present some factorisation and divisibility properties of Catalan polynomials. Initial results established with ad hoc proofs then make way for a more systematic approach and use of the well developed theory of cyclotomic polynomials.

On the evaluation of sums of exponentiated multiples of generalized Catalan number linear combinations using a hypergeometric approach

Larcombe, Peter J. (The Fibonacci Association, 2016-08)

Infinite series comprising exponentiated multiples of p-term linear combinations of Catalan numbers arise naturally from a related power series expansion for sin(2pα) (in odd powers of sin(α)) which itself has an interesting history. In this article some explicit results generated previously by the author (for p = 1, 2, 3) are discussed in the context of this general problem of series summation, and new evaluations made for the cases p = 4, 5 by way of further examples. A powerful hypergeometric approach is adopted which offers, from the analytical formulation developed, a means to achieve these particular evaluations and in principle many others for even greater values of p.

A condition for anti-diagonals product invariance across powers of $2 \times 2$ matrix sets characterizing a particular class of polynomial families

Larcombe, Peter J.; Fennessey, Eric J. (The Fibonacci Association, 2015-05)

Motivated by some recent work on a particular class of polynomial families associated with certain types of integer sequences, we formulate a sufficient condition under which the anti-diagonals products across sets of characterizing 2 × 2 matrices remain invariant as matrix power increases. Two proofs are given along with some examples.

A note on the invariance of the general $2 \times 2$ matrix anti-diagonals ratio with increasing matrix power: Four proofs

Larcombe, Peter J. (The Fibonacci Association, 2015-11)

An invariance matrix property, first observed empirically and seemingly absent from mainstream literature, is stated and established formally here. Four short, and different, proofs are given accordingly.

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