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On small prime divisibility of the Catalan-Larcombe-French sequence
... Consider the integer sequence then again in a more recent article [3] based on the particular expression for the general (n + 1)th term ...
... Accordingly, we present in this note a proof of Theorem 1 which both shortens, and so simplifies, that detailed in [3]. ...
... As seen in [3], in order to establish Theorem 1 it is sufficient merely to show that ...
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 obtain some identities and congruences involving {Sn}. In particular, we determine 7,16, 25, 32, 64, 160, 800, 1600, 156832, where p is an odd prime such that p m. ...
... 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.).
... Clearly, F (n, 1) = 2 n and F (n, 2) = 2n n . The numbers F (n, 3) are called Franel numbers, which have been studied extensively; see, e.g., [3,4] and [9][10][11][12]. ...
Let a,n,m∈ℤ+ with (a,p)=1 and p be an odd prime. We find a supercongruence for ∑p|kapnkm and related sums of powers of binomial coefficients. These results complement prior results for ∑k≡i(mod p)apnkm with i=1,2,…,p−1 obtained recently by the author.
... Recall that in [11], we noted that P p−1 2 is divisible by p if and only if the Franel number f p−1 2 is divisible by p (indeed, they are congruent modulo p). Here, f n = n r=0 n r 3 . ...
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
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
.
It is well known that the numbers (2m)!(2n)!/m!n!(m+n)! are integers,
but in general there is no known combinatorial interpretation for them.
When m=0 these numbers are the middle binomial coefficients C(2n,n), and
when m=1 they are twice the Catalan numbers. In this paper, we give
combinatorial interpretations for these numbers when m=2 or 3.
This sequence was discussed at length in a previous article by Larcombe and French, having arisen originally in a historical formulation by E.C. Catalan. The following short paper presents an analysis of the asymptotic behaviour of the sequence, which also naturally generates the asymptotic form of two associated hypergeometric functions.
The Catalan numbers are are well-known integers that arise in many combinatorial problems. The numbers , , and more generally are also integers for all n. We study the properties of these numbers and of some analogous numbers that generalize the ballot numbers, which we call super ballot numbers.
We consider the topological characteristics of orientable surfaces generated by randomly gluing n triangles together. Our results are most conveniently expressed in terms of a parameter h = n / 2 + χ, where χ is the Euler characteristic of the surface. Simulations and results for similar models suggest that Ex [h] = log(3n) + γ + o(1) and Var [h] = log(3n) + γ - π2 / 6 + o(1). We prove that Ex [h] = log n + O(1) and Var [h] = O(log n). We also derive results concerning a number of other topological invariants and combinatorial characteristics of these random surfaces. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2006
In this paper a very short proof is given of an identity concerning Catalan numbers due originally to Touchard.
Nicholas Pippenger and Kristin Schleich have recently given a combinatorial interpretation for the second-order super-Catalan numbers (u_{n})_{n>=0}=(3,2,3,6,14,36,...): they count "aligned cubic trees" on n internal vertices. Here we give a combinatorial interpretation of the recurrence u_{n} = Sum_{k=0}^{n/2-1} ({n-2}choose{2k} 2^{n-2-2k} u_{k}): it counts these trees by number of deep interior vertices where deep interior means "neither a leaf nor adjacent to a leaf".
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manuscript, 2 pp. [available as the electronic document http://www.lix.polytechnique.fr
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A combinatorial interpretation of super-Catalan numbers of order two, manuscript, 2 pp
- G Schaeffer
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