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On the “other” Catalan numbers: A historical formulation re-examined
... The numbers P n are called Catalan-Larcombe-French numbers since Catalan first defined P n in [2], and in [9] Larcombe and French proved that (1. ...
... The numbers P n occur in the theory of elliptic integrals, and are related to the arithmetic-geometric mean. See [9] and A053175 in Sloane's database "The On-Line Encyclopedia of Integer Sequences". For known properties of P n see also [3,7,8,10]. ...
Let (Formula presented.) be the Catalan–Larcombe–French numbers given by (Formula presented.) and (Formula presented.) (Formula presented.), and let (Formula presented.). In this paper we obtain some identities and congruences involving (Formula presented.). In particular, we determine (Formula presented.) for (Formula presented.), where (Formula presented.) is an odd prime such that (Formula presented.).
... These two sequences came up naturally from the series expansions of the complete elliptic integrals. For more information on {P n } n≥0 and {V n } n≥0 , see [1,3,4,5,6]. ...
... This completes the proof. 4 The monotonicity of { n √ P n } n≥1 and { n √ V n } n≥1 ...
Let denote the Catalan-Larcombe-French sequence, which
naturally came up from the series expansion of the complete elliptic integral
of the first kind. In this paper, we prove the strict log-concavity of the
sequence , which was originally conjectured by
Sun. We also obtain the strict log-concavity of the sequence
, where is the
Fennessey-Larcombe-French sequence arising in the series expansion of the
complete elliptic integral of the second kind.
... The numbers P n are called Catalan-Larcombe-French numbers since Catalan first defined P n in [C], and in [LF1] Larcombe and French proved that ...
... where [x] is the greatest integer not exceeding x. The numbers P n occur in the theory of elliptic integrals, and are related to the arithmetic-geometric-mean. See [LF1] and A053175 in Sloane's database "The On-Line Encyclopedia of Integer Sequences". Let {S n } be defined by (1.3) S 0 = 1, S 1 = 4 and (n + 1) 2 S n+1 = 4(3n 2 + 3n + 1)S n − 32n 2 S n−1 (n ≥ 1). ...
Let be the Catalan-Larcombe-French numbers given by
and , and let
. In this paper we determine
, where p is an odd prime, n is a
positive integer and is the unique nonnegative integer
such that and . We also determine
.
... The Fennessey-Larcombe-French sequence is closely related to the Catalan-Larcombe-French sequence, which was first studied by E. Catalan [1] and later examined and clarified by Larcombe and French [7]. Let {P n } n≥0 denote the Catalan-Larcombe-French sequence, and the following three-term recurrence relation holds: ...
... Many interesting properties have been found for the Catalan-Larcombe-French sequence and the Fennessey-Larcombe-French sequence, and the reader may consult references [5,6,7,8,9,15]. ...
We prove the log-concavity of the Fennessey-Larcombe-French sequence based on
its three-term recurrence relation, which was recently conjectured by Zhao. The
key ingredient of our approach is a sufficient condition for log-concavity of a
sequence subject to certain three-term recurrence.
... Using (2), the asymptotic form of n P was then found [2], at which point the sequence (1) became known as the Catalan-Larcombe-French sequence (Sequence No. A053175 on Sloane's O.E.I.S.). 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. ...
... is integer in order that n P is too. This has been established by Larcombe and French in [1] by four different methods (see Proofs I-IV for , 0 , > b a pp. 50-55) and, embellishing slightly some work by Gessel, also in [3] (see Results 1-3 and the discussion thereof, pp. ...
... 1, 2, 3, 5, 6, 7, 10, 11,13,14,15,17,19,21,22,23,26,29,30. decreasing, and the sequence ( n+1 S(n + 1)/ n S(n)) n 7 is strictly increasing, where S(n) = n k=1 s k . ...
... The Catalan-Larcombe-French numbers P 0 , P 1 , P 2 , . . . (cf. [16]) are given by they arose from the theory of elliptic integrals (see [11]). It is known that (n + 1)P n+1 = (24n(n + 1) + 8)P n − 128n 2 P n−1 for all n ∈ Z + . ...
We pose 30 new conjectures on arithmetical sequences, most of which are about
monotonicity of sequences of the form a_n^{1/n} (n=1,2,...) or the form
a_{n+1}^{1/(n+1)}/a_n^{1/n} (n=1,2,...), where a_n (n=1,2,...) is a
number-theoretic or combinatorial sequence of positive integers. This material
might stimulate further research.
... Larcombe and French [12] give a detailed account of properties of the P n , and obtained [12, Equations (23) and (35)] the following formulas for these numbers: ...
... These numbers occur in the theory of elliptic integrals [12], and there are relations to the arithmetic-geometric-mean [10]. The first few P n are 1, 8, 80, 896, 10816, 137728. ...
We develop the Stienstra-Beukers theory of supercongruences in the setting of the Catalan-Larcombe-French sequence. We also give some applications to other sequences. Comment: 14 pages: v2 has improved formatting
... (n + 1) 2 P (n + 1) = 8(3n 2 + 3n + 1)P (n) − 128n 2 P (n − 1), ∀n 1. P (n) is the 'other' Catalan number which can be given by elliptic integrals (see [LF1]). We will show that (P (n)) ∞ n=1 is realizable (Remark 1.16), while the sequence (C(n)) ∞ n=1 of (true) Catalan numbers is even not almost realizable (Theorem 1.20 (1)). ...
A sequence of non-negative integers is called realizable if there is a map T on a set such that is equal to the number of periodic points of T of (not necessarily exact) period n for all . In this article, we show that for every with , the sequence is realizable, where and . 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.
... which arose from the theory of elliptic integrals (see [11]). It is known that (n + 1)P n+1 = (24n(n + 1) + 8)P n − 128n 2 P n−1 for all n ∈ Z + . ...
In this paper, we mainly prove a congruence conjecture of M. Apagodu [3] and a supercongruence conjecture of Z.-W. Sun [25].
... b; c/ D . 12; 4; 32/ have been studied by Larcombe and French [204] and are sometimes called the Catalan-Larcombe-French numbers. The binomial sums for s.n/ in Table 5.1 were given by Almkvist et al, op. ...
This chapter contains a detailed study of Ramanujan’s theta functions \phi (q) =\sum _{ j=-\infty }^{\infty }q^{j^{2} },\quad \mbox{ and}\quad \psi (q) =\sum _{ j=0}^{\infty }q^{j(j+1)/2}, and the Borweins’ theta functions \displaystyle\begin{array}{rcl} a(q)& =& \sum _{j=-\infty }^{\infty }\sum _{ k=-\infty }^{\infty }q^{j^{2}+jk+k^{2} }, {}\\ b(q)& =& \sum _{j=-\infty }^{\infty }\sum _{ k=-\infty }^{\infty }\omega ^{j-k}q^{j^{2}+jk+k^{2} },\quad \omega =\exp (2\pi i/3), {}\\ \mbox{ and}\quad c(q)& =& \sum _{j=-\infty }^{\infty }\sum _{ k=-\infty }^{\infty }q^{(j+\frac{1} {3} )^{2}+(j+\frac{1} {3} )(k+\frac{1} {3} )+(k+\frac{1} {3} )^{2} }. {}\\ \end{array}
... b; c/ D . 12; 4; 32/ have been studied by Larcombe and French [204] and are sometimes called the Catalan-Larcombe-French numbers. The binomial sums for s.n/ in Table 5.1 were given by Almkvist et al, op. ...
We prove that and obtain results that are analogues of theorems in Chapters 5, 6, and 7.
... b; c/ D . 12; 4; 32/ have been studied by Larcombe and French [204] and are sometimes called the Catalan-Larcombe-French numbers. The binomial sums for s.n/ in Table 5.1 were given by Almkvist et al, op. ...
We present a variety of techniques for analyzing some remarkable series for 1∕π, such as that were first studied by Ramanujan. It is shown how to classify the series by level and degree. We also obtain iterative processes that converge rapidly to 1∕π.
... b; c/ D . 12; 4; 32/ have been studied by Larcombe and French [204] and are sometimes called the Catalan-Larcombe-French numbers. The binomial sums for s.n/ in Table 5.1 were given by Almkvist et al, op. ...
The results in this chapter are basic tools that will be used throughout the book. They include the fundamental identities for theta functions such as Jacobi’s triple product identity, the quintuple product identity, Ramanujan’s summation formula, and the q-binomial theorem. We also encounter generalizations of the sine and cosine functions. A study of the coefficients in their power series expansions leads to a system of nonlinear differential equations, called Ramanujan’s differential equations.
... The sequences {P n } n≥0 and {V n } n≥0 are known as the Catalan-Larcombe-French sequence and the Fennessey-Larcombe-French sequence, respectively. They arise naturally from the series expansions of the complete elliptic integrals, see [2,6,7,8,9]. The main objective of this paper is to prove the ratio log-concavity of {P n } n≥0 , the log-convexity of {V 2 n −V n−1 V n+1 } n≥2 , the ratio log-convexity of {V n } n≥1 , and the log-convexity of {n!V n } n≥1 . ...
Two interesting sequences arose in the study of the series expansions of the complete elliptic integrals, which are called the Catalan-Larcombe-French sequence {Pn}n≥0 and the Fennessey-Larcombe-French sequence {Vn}n>0 respectively. In this paper, we first establish some criteria for determining log-behavior of a sequence based on its three-term recurrence. Then we prove the log-convexity of {V2n
− Vn−1Vn+1}n≥2 and {n!Vn}n≥1, the ratio log-concavity of {Pn}n≥0 and the sequence {An}n≥0 of Apéry numbers, and the ratio log-convexity of {Vn}n≥1.
... Note that these numbers were also discovered independently by Segner (Larcombe and French, 2000) and Majdalani (2009), the latter expressed them as with the above data, a recursive relation is formulated ...
... In their delightful paper, Larcombe and French [3] developed a number of properties of the sequence (A053175) P 0 = 1, P 1 = 8, P 2 = 80, P 3 = 896, P 4 = 10816, . . . originally discussed by Catalan [1]. ...
We give an elementary development of a complete asymptotic expansion for the Catalan-Larcombe-French sequence.
We prove that and develop the rich properties of the infinite product.
The Catalan-Larcombe-French sequence arises in a series expansion of the complete elliptic integral of the first kind. It has been proved that the sequence is log-balanced. In the paper, by exploring a criterion due to Chen and Xia for testing 2-log-convexity of a sequence satisfying three-term recurrence relation, we prove that the new sequence are strictly log-convex and hence the Catalan-Larcombe-French sequence is strictly 2-log-convex.
The harmonic numbers H_n=\sum_{0 play
important roles in mathematics. With helps of some combinatorial identities, we
establish the following two congruences:
\sum_{k=0}^{\frac{p-3}2}\f{\binom{2k}k^2H_k}{(2k+1)16^k}\ \mbox{modulo}\ p^2\
\mbox{and}\ \sum_{k=0}^{\frac{p-3}2}\f{\binom{2k}k^2H_{2k}}{(2k+1)16^k}\
\mbox{modulo}\ p for any prime , the second one was conjectured by Z.-W.
Sun in 2012. These two congruences are very important to prove the following
conjectures of Z.W.Sun: For any old prime p, we have
and
where is
the n-th Catalan-Larcombe-French number.
We present a short proof of the 2-adic valuation of the general term of the Catalan-Larcombe-French sequence which, following its initial dissemination by Eugène Catalan in the late 19th century, has been recovered formally in two different ways in previous papers.
In this paper, we discuss the log-behavior of the Catalan–Larcombe–French sequence {Pn}n≥0. We prove that {Pn}n≥0 is log-balanced and {Pn/(n!)2}n≥0 is unimodal. In addition, we show that {Pk/k!}0≤k≤n is reverse ultra log-concave.
Let f be a modular form of weight k for a congruence subgroup Γ ⊂ SL2(Z), and t a weight 0 modular function for Γ. Assume that near t = 0, we can write f = σn<0bn t n, bn ∈ Z. Let ℓ(z) be a weight k + 2 modular form with q-expansion σγnqn, such that the Mellin transform of ℓ can be expressed as an Euler product. Then we show that if fq/t dt/dq = σ ai(diz) for some integers ai, di, then the congruence relation bmpr -γpbmpr-1 + εppk+1b mpr-2 ≡ 0 (mod pr) holds. We give a number of examples of this phenomena.
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