An Asymptotic Expansion for the Catalan-Larcombe-French Sequence

Lane Clark Department of Mathematics Southern Illinois University, Carbondale Carbondale, IL 62901–4408 USA

                      lclark@math.siu.edu

Abstract We give an elementary development of a complete asymptotic expansion for the

Catalan-Larcombe-French sequence.

1 Introduction

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]. In addition to a generating function, the following formula for P_{n} was derived

P_{n} = \frac{1}{n!} ∑_{p+q=n} \binom{2p}{p} \binom{2q}{q} \frac{(2p)! (2q)!}{p! q!}  (n ∈ ℕ) .  (1)

Recently, Larcombe et al. [4] showed that P_{n}/2\binom{2n}{n}^{2} → 1 as n → ∞ by a rather lengthy analysis. In this short paper, we give an elementary development of a complete asymptotic expansion for P_{n}/2\binom{2n}{n}^{2}. We conclude with a table of numerical calculations as a companion of the theoretical results.

2 Main result

The positive integers are denoted by ℙ; the nonnegative integers by ℕ; the nonnegative rational numbers by ℚ_{0}; and the complex numbers by ℂ. Let z^{0} = (z)_{0} ≡ 1 and (z)_{p} = (z) ⋯ (z − p + 1) when z ∈ ℂ and p ∈ ℙ. For z ∈ ℂ and p ∈ ℙ, (z)_{p} = ∑_{q=0}^{p} s(p, q)z^{q} where the s(p, q) are the Stirling numbers of the first kind (see [2; pp. 212–214]). We write f(z) = O(g(z)) provided there exist real constants C, D with |f(z)| ≤ C|g(z)| for all |z| ≥ D.

For 0 ≤ p ≤ n, let

a_{p} = \binom{2p}{p}^{2} \binom{2n − 2p}{n − p}^{2}p! (n − p)! ∈ ℙ

hence, a_{p} = a_{n−p} and, let

b_{p} = \binom{2p}{p} \binom{2n − 2p}{n − p} ∈ ℙ

hence, b_{p} = b_{n−p} and a_{p} = n! b_{p}^{2}\binom{n}{p}^{−1}. For 0 ≤ p ≤ n − 1,

\frac{b_{p+1}}{b_{p}} = \frac{(2p + 1)(n − p)}{(p + 1)(2n − 2p + 1)}

then, for 0 ≤ p ≤ (n − 1)/2,

0 < \frac{b_{p+1}}{b_{p}} ≤ 1

hence, for 1 ≤ p ≤ (n − 1)/2

0 < \frac{b_{p}}{b_{0}} = \frac{b_{p}}{b_{p−1}} ⋯ \frac{b_{1}}{b_{0}} ≤ 1  (2)

which is correct for p = 0 also.

Fix s ≥ 1. For n = 2m + 1 ≥ 2s + 3, (1) and symmetry give

P_{n} = \frac{1}{n!} ∑_{p=0}^{n} \binom{2p}{p} \binom{2n − 2p}{n − p} \frac{(2p)! (2n − 2p)!}{p! (n − p)!}

= \frac{1}{n!} ∑_{p=0}^{n} a_{p} = \frac{2}{n!} ∑_{p=0}^{m} a_{p} = 2 ∑_{p=0}^{m} b_{p}^{2}\binom{n}{p}^{−1}

= 2b_{0}^{2} \{∑_{p=0}^{s} (\frac{b_{p}}{b_{0}} )^{2} \binom{n}{p}^{−1} + ∑_{p=s+1}^{m} (\frac{b_{p}}{b_{0}} )^{2} \binom{n}{p}^{−1}\} .  (3)

Now \binom{n}{p} is an increasing sequence of positive integers for 0 ≤ p ≤ (n − 1)/2, so (2) gives

0 < ∑_{p=s+1}^{m} (\frac{b_{p}}{b_{0}} )^{2} \binom{n}{p}^{−1} ≤ n\binom{n}{s + 1}^{−1} = O(n^{−s}) as n → ∞.  (4)

For 0 ≤ p ≤ (n − 1)/2,

\frac{b_{p}}{b_{0}} = \binom{2p}{p} \frac{(n)_{p}^{2}}{(2n)_{2p}}

hence,

∑_{p=0}^{s} (\frac{b_{p}}{b_{0}} )^{2} \binom{n}{p}^{−1} = ∑_{p=0}^{s} p! \binom{2p}{p}^{2} \frac{(n)_{p}^{3}}{(2n)_{2p}^{2}} .  (5)

For z ∈ ℂ, let f_{0}(z) ≡ 1 and, for 1 ≤ p ≤ s, let

f_{p}(z) = p!\binom{2p}{p}^{2} \frac{(z)_{p}^{3}}{(2z)_{2p}^{2}} = \frac{p! \binom{2p}{p}^{2}(z)_{p}}{2^{4p}z^{2p}} ∏_{j=1}^{p} (1 − \frac{2j − 1}{2z} )^{−2}

= \frac{p! \binom{2p}{p}^{2}}{2^{4p}} ∑_{q=0}^{p} s(p, q)z^{q−2p} \{∏_{j=1}^{p} (∑_{r=0}^{∞}(r + 1)(j − 0.5)^{r}z^{−r})\}

= ∑_{r=p}^{∞} b(p, r)z^{−r}  (z ∈ ℂ; |z| ≥ s)  (6)

where b(p, r) ∈ ℚ_{0} for r ≥ p and b(p, p) = p! \binom{2p}{p}^{2}/2^{4p}. For 1 ≤ p ≤ s,

|(z)_{p}| ≤ (|z| + s)^{p} ≤ (\frac{3}{2}|z|)^{p}
(|z| ≥ 2s)
|(2z)_{2p}| ≥ (2|z| − 2s)^{2p} ≥ |z|^{2p} ,

hence,

|\frac{(z)_{p}^{3}}{(2z)_{2p}^{2}} | ≤ (\frac{4}{|z|})^{p}  (|z| ≥ 2s).  (7)

For r, s ≥ p ≥ 1, Laurent’s Theorem (see [5; V. 2, p. 6]), standard estimates for the integral and (7) give

|b(p, r)| = | \frac{1}{2πi} ∮_{|z|=2s} \frac{f_{p}(z)}{z^{r+1}} dz |

≤ p! \binom{2p}{p}^{2} (\frac{2}{s})^{p} \frac{1}{(2s)^{r}} ≤ s! \binom{2s}{s}^{2} .  (8)

Then (8) gives

|∑_{r=s}^{∞} b(p, r)z^{−r}| ≤ |z|^{−s}s! \binom{2s}{s}^{2} ∑_{t=0}^{∞} |z|^{−t} = O(|z|^{−s}) ,  (9)

hence, (6,9) give

f_{p}(z) = ∑_{r=p}^{s−1} b(p, r)z^{−r} + O(|z|^{−s})  (10)

where f_{s}(z) = O(|z|^{−s}). For s ≥ 1, (10) gives

g_{s}(z) := ∑_{p=0}^{s−1} f_{p}(z)  (z ∈ ℂ; |z| ≥ s)

= ∑_{r=0}^{s−1} c(s, r)z^{−r} + O(|z|^{−s})  (11)

where c(s, 0) = 1 for s ≥ 1 and c(s, r) = ∑_{p=1}^{r} b(p, r) ∈ ℚ_{0} for 1 ≤ r ≤ s − 1. Observe that c(s + 1, r) = c(s, r) for 0 ≤ r ≤ s − 1. The analysis for n = 2m ≥ 2s + 2 is identical except that P_{n} includes b_{m}^{2}\binom{n}{m}^{−1} not 2b_{m}^{2}\binom{n}{m}^{−1}. Then (3–5,11) give the following complete asymptotic expansion for P_{n}/2\binom{2n}{n}^{2}.

Theorem. Fix s ≥ 1. There exist effectively calculable nonnegative rational numbers c(s, 0) = 1, c(s, 1), …, c(s, s − 1) so that

P_{n}/2\binom{2n}{n}^{2} = ∑_{r=0}^{s−1} c(s, r)n^{−r} + O(n^{−s}) as n → ∞. ■

Let

h_{s}(n) = ∑_{r=0}^{s−1} c(s, r)n^{−r}

hence,

h_{1}(n) = 1, h_{2}(n) = 1 + \frac{1}{4n} , h_{3}(n) = 1 + \frac{1}{4n} + \frac{17}{32n^{2}} ,
h_{4}(n) = 1 + \frac{1}{4n} + \frac{17}{32n^{2}} + \frac{207}{128n^{3}} and
h_{5}(n) = 1 + \frac{1}{4n} + \frac{17}{32n^{2}} + \frac{207}{128n^{3}} + \frac{14875}{2048n^{4}} .

Let Q_{s}(n) = [P_{n}/2\binom{2n}{n}^{2} −h_{s}(n)]n^{s}. Our theorem shows Q_{s}(n) = O(1) as n → ∞. The table below which gives the first 10 digits of Q_{2}(n) and Q_{3}(n) for several values of n was found using Mathematica. This provides numerical evidence for our theorem.

\begin{matrix}n & Q_{2}(n) & Q_{3}(n) \\ 100 & .5481946735 & 1.6944673581 \\ 200 & .5395231003 & 1.6546200668 \\ 300 & .5367229603 & 1.6418880975 \\ 400 & .5353390483 & 1.6356193435 \\ 500 & .5345137769 & 1.6318884961 \\ 600 & .5339656896 & 1.6294137740 \\ 700 & .5335752174 & 1.6276521934 \\ 800 & .5332829178 & 1.6263343131 \\ 900 & .5330559013 & 1.6253112456 \\ 1000 & .5328744940 & 1.6244940166 \\ 2000 & .5320604149 & 1.6208298850 \\ 3000 & .5317898711 & 1.6196133523 \end{matrix}

Acknowledgement. I wish to thank the referee for comments and suggestions which have resulted in an improved version of the paper.

References

1. E. Catalan, Sur les Nombres de Segner, Rend. Circ. Mat. Pal. 1 (1887), 190–201.

2. L. Comtet, Advanced Combinatorics, D. Reidel, Boston, MA, 1974.

3. P.J. Larcombe and D.R. French, On the ‘Other’ Catalan Numbers: A Historical Formulation Re-examined, Congr. Numer. 143 (2000), 33–64.

4. P.J. Larcombe, D.R. French and E.J. Fennessey, The Asymptotic Behaviour of the Catalan-Larcombe-French Sequence {1, 8, 80, 896, 10816, …}, Util. Math. 60 (2001), 67–77.

5. A.I. Markushevich, Theory of Functions of a Complex Variable, Chelsea Publishing Co., New York, NY, 1985.

2000 Mathematics Subject Classification: Primary 05A10; Secondary 05A16, 41A60.

Keywords: asymptotic expansion, Catalan-Larcombe-French sequence.

(Concerned with sequence \underline{A053175}.)

Received October 28 2002; revised version received February 2 2004. Published in Journal of Integer Sequences, March 23 2004.

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