Article

A new generating function for the Catalan-Larcombe-French sequence: proof of a result by Jovovic

Authors:
Peter J. Larcombe
Peter J. Larcombe
Peter J. Larcombe
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David R. French
David R. French
David R. French
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... In [9] a generating function for the numbers P n is given in terms of the square of a modified Bessel function, and we use this approach to obtain an asymptotic expansion of f (n). See also [7] for details on this generating function. ...
... I wish to thank Peter Larcombe for suggesting this problem, for encouraging me to investigate the asymptotic properties of the sequence {P n }, and for introducing me to the literature, in particular to the papers [4] - [7]. ...
Asymptotics of a 3F2 polynomial associated with the Catalan-Larcombe-French sequence
Article
Full-text available
  • Oct 2006
The large n behavior of the hypergeometric polynomial [Formula: see text] is considered by using integral representations of this polynomial. This 3 F 2 polynomial is associated with the Catalan–Larcombe–French sequence. Several other representations are mentioned, with references to the literature, and another asymptotic method is described by using a generating function of the sequence. The results are similar to those obtained by Clark (2004) who used a binomial sum for obtaining an asymptotic expansion.
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... In particular, The sequence of Domb numbers and the sequence of Zaiger numbers are both realizable. Moreover, the sequence of Catalan-Larcombe-French numbers is also realizable: [LF2,Theorem 3] implies that P (n) = 2 n Z(n) for all n 0. As in Example 1.4 (3),the sequence (2 n ) ∞ n=1 is realized via the shift map T : ...
Realizability of Some Combinatorial Sequences
Preprint
  • Feb 2023
A sequence a=(an)n=1∞a=(a_n)_{n=1}^\infty of non-negative integers is called realizable if there is a map T on a set such that ana_n is equal to the number of periodic points of T of (not necessarily exact) period n for all n⩾1n\geqslant1. In this article, we show that for every r1,r2,s,t,u∈Z⩾0r_1,r_2,s,t,u\in\mathbb{Z}_{\geqslant0} with r1+r2⩾1r_1+r_2\geqslant1, the sequence (V(n,r1,r2,s,t,u))n=1∞(V(n,r_1,r_2,s,t,u))_{n=1}^\infty is realizable, where V(n,r1,r2,s,t,u)=∑k=0n(nk)r1(n2k)r2(n+kk)s(2kk)t(2(n−k)n−k)uV(n,r_1,r_2,s,t,u)=\sum_{k=0}^n \binom{n}{ k}^{r_1}\binom{n}{2k}^{r_2}\binom{n+k}{k}^s\binom{2k}{k}^t\binom{2(n-k)}{n-k}^u and 00=10^0=1. From this, we deduce that many famous combinatorial sequences are realizable, for example, the sequences of certain Ap\'ery-like numbers, Franel numbers of any order and central trinomial coefficients; while we also show that the sequences of the Catalan numbers, Motzkin numbers and Schr\"oder numbers are even not almost realizable.
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... In this paper we investigate the properties of S n instead of P n since S n is an Apépy-like sequence. As observed by V. Jovovic in 2003 (see [LF2]), ...
Congruences for Catalan-Larcombe-French numbers
Article
  • May 2015
  • PUBL MATH-DEBRECEN
Let {Pn}\{P_n\} be the Catalan-Larcombe-French numbers given by P0=1, P1=8P_0=1,\ P_1=8 and n2Pn=8(3n2−3n+1)Pn−1−128(n−1)2Pn−2n^2P_n=8(3n^2-3n+1)P_{n-1}-128(n-1)^2P_{n-2} (n≥2)(n\ge 2), and let Sn=Pn/2nS_n=P_n/2^n. In this paper we determine Snp−Sn(modp3+ordpn)S_{np}-S_n\pmod{p^{3+\text{ord}_pn}}, where p is an odd prime, n is a positive integer and ordpn\text{ord}_pn is the unique nonnegative integer α\alpha such that pα∣np^{\alpha}\mid n and pα+1∤np^{\alpha+1}\nmid n. We also determine Snp+1(modp3)S_{np+1}\pmod{p^3}.
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... Regarding the integer nature of the general term n P , this was discussed at some length in [1] and subsequently, having re-visited the topic, it was shown formally in [3, pp. 81-83] that for Equation (2) shows by inspection the positivity of n P but not integrality, whilst (5), (6) give both. Equation (3), on the other hand, shows integrality but not positivity, and it is on this (the most convenient) form of n P that we base our study of its divisibility by some small primes. ...
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On rationally parametrized modular equations
Article
Full-text available
  • Nov 2006
Many rationally parametrized elliptic modular equations are derived. Each comes from a family of elliptic curves attached to a genus-zero congruence subgroup Γ0(N)\Gamma_0(N), as an algebraic transformation of elliptic curve periods, parametrized by a Hauptmodul (function field generator). The periods satisfy a Picard-Fuchs equation, of hypergeometric, Heun, or more general type; so the new modular equations are algebraic transformations of special functions. When N=4,3,2 they are modular transformations of Ramanujan's elliptic integrals of signatures 2,3,4. This gives a modern interpretation to his theories of integrals to alternative bases: they are attached to certain families of elliptic curves. His anomalous theory of signature 6 turns out to fit into a general Gauss-Manin rather than a Picard-Fuchs framework. Comment: 57 pages, 19 tables
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