A criterion for the log-convexity of combinatorial sequences Ernest X. W. Xia
∗
Olivia X. M. Yao
Department of Mathematics Jiangsu University Zhenjiang, Jiangsu, P. R. China
Department of Mathematics Jiangsu University Zhenjiang, Jiangsu, P. R. China
[email protected] [email protected] Submitted: May 28, 2013 ; Accepted: Sep 30, 2013; Published: Oct 14, 2013 Mathematics Subject Classifications: 05A20, 11B83
Abstract Recently, Doˇsli´c, and Liu and Wang developed techniques for dealing with the log-convexity of sequences. In this paper, we present a criterion for the log-convexity of some combinatorial sequences. In order to prove the log-convexity of a sequence satisfying a three-term recurrence, by our method, it suffices to compute a constant number of terms at the beginning of the sequence. For example, in order to prove the log-convexity of the Ap´ery numbers An , by our method, we just need to evaluate the values of An for 0 6 n 6 6. As applications, we prove the log-convexity of some famous sequences including the Catalan-Larcombe-French numbers. This confirms a conjecture given by Sun. Keywords: log-convexity; three-term recurrence; combinatorial sequences
1
Introduction
A positive sequence {Sn }∞ n=0 is said to be log-convex (respectively log-concave) if for n > 1, Sn Sn+1 6 Sn−1 Sn
(respectively
Sn Sn+1 > ). Sn−1 Sn
(1)
Meanwhile, the sequence {Sn }∞ n=0 is called strictly log-convex (log-concave) if the inequality in (1.1) is strict for all n > 1. In 1994, Engel [8] proved the log-convexity of the Bell numbers. Recently, Doˇsli´c [4, 5, 6], Doˇsli´c and Veljan [7], and Liu and Wang [14] developed techniques for proving the log-convexity of sequences. Doˇsli´c [4, 5, 6] presented ∗
Supported by the National Natural Science Foundation of China (11201188).
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several methods for dealing with the log-convexity of combinatorial sequences. He proved that the Motzkin numbers, the Fine numbers, the Franel numbers of order 3 and 4, the Ap´ery numbers, the large Schr¨oder numbers, the derangements numbers and the central Delannoy numbers are log-convex. In their wonderful paper [14], Liu and Wang proved that the log-convexity is preserved under componentwise sum, under binomial convolution, and by the linear transformations given by the matrices of binomial coefficients and Stirling numbers of two kinds. Many combinatorial sequences satisfy a three-term recurrence. Liu and Wang [14] presented some criterions for the log-convexity of the sequences {zn }∞ n=0 satisfying the following recurrence a(n)zn+1 = b(n)zn + c(n)zn−1 ,
(2)
where a(n), b(n) and c(n) are positive for n > 1, Liu and Wang [14] proved the following theorem. Theorem 1. Let {zn }∞ n=0 be defined by (2) and p b(n) + b2 (n) + 4a(n)c(n) . λn = 2a(n)
(3)
Suppose that z0 , z1 , z2 , z3 is log-convex and that the inequality a(n)λn−1 λn+1 − b(n)λ(n − 1) − c(n) > 0
(4)
is true for n > 2. Then the sequence {zn }∞ n=0 is log-convex. Liu and Wang [14] also considered the log-convexity of the sequence {zn }∞ n=0 defined by (αn + α0 )zn+1 = (β1 n + β0 )zn − (γ1 n + γ0 )zn−1 .
(5)
for n > 1. They gave criterions for the log-convexity of the sequences {zn }∞ n=0 . Employing their criterions, they proved the log-convexity of some combinatorial sequences. Liu [13] gave sufficient conditions for the positivity of the sequences defined by (5). Motivated by these results established by Liu and Wang [14], in this paper, we investigate the log-convexity problem of the sequence {Sn }∞ n=0 having the following three-term recurrence Pk Pl i ci ni i=0 ai n Sn = Pk Sn−1 − Pli=0 Sn−2 (n > 2), (6) i i b n d n i i i=0 i=0 P P P P where gcd( ki=0 ai ni , ki=0 bi ni ) = gcd( li=0 ci ni , li=0 di ni ) = 1 and k, l, ak , bk , cl and dl are positive numbers. The authors [19] gave a criterion for the positivity of the sequence {Sn }∞ n=0 defined by (6). The aim of this paper is to present a criterion for the logconvexity of some famous combinatorial sequences. By our method, in order to determine the log-convexity of the sequence {Sn }∞ n=0 defined by (6), it suffices to compute a constant the electronic journal of combinatorics 20(4) (2013), #P3
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number of terms at the beginning of the sequence {Sn }∞ n=0 . As applications, we prove some famous combinatorial sequences are strictly log-convex. Specially, we show that the Catalan-Larcombe-French numbers {Pn }∞ n=0 is strictly log-convex which confirms a conjecture given by Sun [18]. In order to state our main result, we first introduce some notations. Given a polynomial f (n) defined by f (n) =
k X
f i ni ,
(7)
i=0
where fi (0 6 i 6 k) are real numbers and fk > 0. Define an operator L on f (n) by L(f (n)) =
1 fk
X
|fi |.
(8)
06i6k−1, fi 0 for n > [L(f (n))] + 1. Throughout this paper, we always let Pr Pk Pk j i i a (n + 2) a (n + 1) i i j=0 ej n i=0 i=0 , − Pk = Ps Pk t i i t=0 ht n i=0 bi (n + 2) i=0 bi (n + 1) Pu Pl Pl j i i j=0 pj n i=0 ci (n + 2) i=0 ci (n + 1) P − = , Pl Pl v t i i q n t d (n + 2) d (n + 1) t=0 i i i=0 i=0
(10)
(11)
and P Pv r j t i ( t=0 qt n ) j=0 ej n ai (n + 2)i i=0 ci (n + 2) − Pi=0 P Ps k l i i Pu t j ( i=0 bi (n + 2) i=0 di (n + 2) t=0 ht n ) j=0 pj n
Pk
Pl
P
P Pα pj (n + 1) ( st=0 ht (n + 1)t ) xi ni P = Pi=0 − P , β i r j y in ( vt=0 qt (n + 1)t ) e (n + 1) i=0 j j=0 u j=0
j
(12)
where hs > 0, qv > 0, yβ > 0 and ! ! ! β r s u v α X X X X X X j t j t i i gcd ej n , ht n = gcd pj n , qt n = gcd xi n , yi n = 1. j=0
t=0
j=0
t=0
i=0
i=0
Our main result can be stated as follows. the electronic journal of combinatorics 20(4) (2013), #P3
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Theorem 2. Let {Sn }∞ n=0 be a positive sequence and satisfy (6). If pu > 0, er > 0, xα > 0 and there exists an integer N0 such that # " # " # " # " X l l r s X X X N0 > r1 = max L( ci ni ) , L( di ni ) , L( ej nj ) , L( ht nt ) , i=0
"
i=0
t=0
j=0
# β u v α X X X X yi ni ) + 1 (13) pj nj ) , L( qt nt ) , L( xi ni ) , L( L( # "
j=0
# "
t=0
# "
i=0
i=0
and SN0 SN0 +1 < , SN0 −1 SN0 P P s u j t t=0 ht N0 ) j=0 pj N0 ( SN0 +1 , P > P r SN0 e Nj ( v q N t) t=0 t
j=0 j
0
(14)
(15)
0
then the sequence {Sn }∞ n=N0 is strictly log-convex, namely, Sn Sn+1 < , Sn−1 Sn
(n > N0 ).
(16)
This paper is organized as follows. We give the proof of Theorem 2 in Sections 2. As applications of Theorem 2, in Section 3, we prove the log-convexity of some famous sequences including the Catalan-Larcombe-French numbers. This confirms a conjecture given by Sun [18].
2
Proof of Theorem 2
In this section, we present the proof of Theorem 2. Proof. By the definition of r1 , we see that for all n > N0 > r1 , Pl ci ni > 0, Pli=0 i i=0 di n Pr Pk Pk j i i a (n + 2) a (n + 1) i i j=0 ej n i=0 i=0 − Pk = Ps > 0, Pk t i i t=0 ht n i=0 bi (n + 2) i=0 bi (n + 1) Pu Pk Pk j i i j=0 pj n i=0 ci (n + 2) i=0 ci (n + 1) P − = > 0, Pk Pk v t i i q n t d (n + 2) d (n + 1) t=0 i i i=0 i=0
(17)
(18)
(19)
and Pα
Pi=0 β
i=0
xi ni yi ni
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> 0.
(20)
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We first give a lower bound for
Sn+1 . Sn
Sn+1 Sn
Moreover, we prove that for n > N0 , P P u j ( st=0 ht nt ) p n j=0 j P . > P r j ( vt=0 qt nt ) e n j=0 j
(21)
We prove (21) by induction on n. By (15), we see that (21) holds for n = N0 . Suppose that (21) holds for n = m > N0 , that is, P P u j p m ( st=0 ht mt ) j=0 j Sm+1 P . > P (22) v r Sm t j ( qm) em t=0 t
j=0 j
It follows from (17), (18) and (22) that for m > N0 , Pl
ci (m + 2)i Sm i Sm+1 i=0 di (m + 2)
− Pli=0
P Pv r j t j=0 ej m ci (m + 2)i ( t=0 qt m ) P P . > − Pli=0 i s u t) j ( h m p m i=0 di (m + 2) t j t=0 j=0 Pl
(23)
Now, we are ready to show that (21) also holds for n = m + 1. Employing (6) and (23), we deduce that Pl Pk i ci (m + 2)i Sm Sm+2 i=0 ai (m + 2) − Pli=0 = Pk i i Sm+1 Sm+1 i=0 bi (m + 2) i=0 di (m + 2) P Pv r Pl Pk j t i ( i e m q m ) j t j=0 t=0 ci (m + 2) ai (m + 2) P P . (24) − Pli=0 > Pi=0 k i i s u t) j ( h m p m i=0 bi (m + 2) i=0 di (m + 2) t=0 t j=0 j In view of (12), (20) and (24), we find that for m > N0 P P u j ( st=0 ht (m + 1)t ) p (m + 1) j=0 j Sm+2 P − P v r Sm+1 t j ( q (m + 1) ) e (m + 1) t=0 t
j=0 j
P Pv r t j i ( t=0 qt m ) j=0 ej m ai (m + 2)i i=0 ci (m + 2) − > Pi=0 P Ps k l i i Pu j ( t i=0 bi (m + 2) i=0 di (m + 2) j=0 pj m t=0 ht m ) Pl
Pk
P u
P ( st=0 ht (m + 1)t ) j=0 pj (m + 1) P − P r j ( vt=0 qt (m + 1)t ) e (m + 1) j j=0 Pα
= Pi=0 β
i=0
xi mi yi mi
j
> 0,
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(25)
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which implies that (21) is true for n = m + 1. By induction, we have proved (21) holds for n > N0 . Now, we turn to prove (16). We also prove (16) by induction on n. It follows from (14) that (16) holds for n = N0 . Assume that (16) is true for n = m > N0 , namely, Sm+1 Sm < . Sm−1 Sm
(26)
By (17) and (26), we find that for m > N0 Pl
i i=0 ci (m + 1) Pl i i=0 di (m + 1)
Pl ci (m + 1)i Sm Sm−1 > Pli=0 . i Sm+1 Sm d (m + 1) i i=0
(27)
Employing (6), (10), (11), (19), (21) and (27), we deduce that for m > N0 , Pk Pl i ci (m + 2)i Sm Sm+2 Sm+1 i=0 ai (m + 2) − = Pk − Pli=0 i i Sm+1 Sm+1 Sm i=0 bi (m + 2) i=0 di (m + 2) Pk Pl i ci (m + 1)i Sm−1 i=0 ai (m + 1) − Pk + Pli=0 i i Sm i=0 bi (m + 1) i=0 di (m + 1) Pk Pk i ai (m + 1)i i=0 ai (m + 2) − Pi=0 > Pk k i i i=0 bi (m + 2) i=0 bi (m + 1) ! Pl Pl i i c (m + 1) c (m + 2) Sm i i + Pli=0 − Pli=0 i i Sm+1 i=0 di (m + 1) i=0 di (m + 2) Pu Pr j j Sm j=0 pj m j=0 ej m − Pv = Ps t t t=0 ht m t=0 qt m Sm+1 P Pv Pu Pr r j t j j ( t=0 qt m ) j=0 ej m j=0 pj m j=0 ej m = 0, P − > Ps v Ps Pu t t h m q m t j t t t=0 t=0 ( t=0 ht m ) j=0 pj m
(28)
which implies that (16) holds for n = m + 1. Theorem 2 is proved by induction. This completes the proof.
3
Applications of Theorem 2
In this section, employing the criterion given in this paper, we prove some results on the log-convexity of some combinatorial sequences. The Catalan-Larcombe-French numbers Pn for n > 0 were first defined by Catalan in [2], in terms of the “Segner numbers”. Catalan stated that the Pn could be defined by
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the following recurrence relation: 8(3n2 − 3n + 1) 128(n − 1)2 Pn = Pn−1 − Pn−2 , n2 n2
(29)
for n > 2, with the initial values given by P0 = 1 and P1 = 8. Larcombe and French [12] gave a detailed account of properties of Pn , and obtained the following formulas for these numbers: bn/2c n 2k 2 2(n−k) 2 X n 2k 2 X k n−k n 4n−2k (30) =2 Pn = n 2k k k k=0 k=0 and X 1 X 2r 2s (2r)!(2s)! Pn = = n! r+s=n r s r!s! r+s=n
2r 2 2s 2 r s n r
(31)
for n > 0. The first few Pn are 1, 8, 80, 896, 10816, 137728. This is the sequence A053175 in Sloane’s database [16]. The sequence {Pn }∞ n=0 is also related to the theory of modular forms; see [20]. Recently, Sun [18] conjectured that √ n ∞ Conjecture 3. The sequences {Pn+1 /Pn }∞ n=0 and { Pn }n=1 are strictly increasing. Employing Theorem 2, we prove that Corollary 4. Conjecture 3 is true. Proof. By (13), we find r1 = 3. Set N0 = 3. It is easy to check that (14) and (15) hold for N0 = 3. By Theorem 2, we see that the sequence {Pn }∞ n=3 is strictly log-convex. It is Pi+1 Pi a routine to verify that Pi > Pi−1 for 1 6 i 6 3. Thus, the sequence {Pn }∞ n=0 is strictly ∞ log-convex and the sequence {Pn+1 /Pn }n=0 is strictly increasing, namely, Pn+1 Pn > , Pn Pn−1
n > 1.
(32)
By (32) and the fact P0 = 1, we deduce that n n Y Pi Pn+1 Pn = P0 < , P Pn i=1 i−1
(33)
which implies that n Pnn+1 < Pn+1 . (34) √ It follows from (34) that the sequences { n Pn }∞ n=1 is strictly increasing. This completes the proof.
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The Ap´ery number An is defined by An =
34n3 − 51n2 + 27n − 5 (n − 1)3 A − An−2 , n−1 n3 n3
n > 2,
(35)
with A0 = 1 and A1 = P 5. The Ap´ery numbers play a key role in Ap´ery’s proof of the ∞ 1 ∞ irrationality of ζ(3) = n=1 n3 ; see [1]. The log-convexity of {An }n=0 was proved by Doˇsli´c [4]. Chen and Xia [3] proved that the sequence {An }∞ n=0 is 2-log-convex. Now, ∞ we present another proof of the log-convexity of {An }n=0 . Set k = l = 3, a3 = 34 and b3 = c3 = d3 = 1 in Theorem 2. By the definition of r1 , we obtain r1 = 5. Set N0 = 5. We can check that (14) and (15) hold for N0 = 5. Thus, by Theorem 2, the sequence Ai+1 Ai {An }∞ n=5 is log-convex. We can also verify that Ai > Ai−1 for 1 6 i 6 5. Thus, the following corollary is true. Corollary 5. The sequence {An }∞ n=0 is strictly log-convex. The central Delannoy number Dn is defined by Dn =
n−1 3(2n − 1) Dn−1 − Dn−2 , n n
n > 2,
(36)
with D0 = 1 and D1 = 3; see [15]. Doˇsli´c [4] , and Liu and Wang [14] proved the logconvexity of the sequence {Dn }∞ n=0 . By (13), we find r1 = 2. Let N0 = 2. It is easy to check that (14) and (15) hold for N0 = 2. The following corollary follows from Theorem D2 D1 2 and the fact D >D . 1 0 Corollary 6. The sequence {Dn }∞ n=0 is strictly log-convex. The little Schr¨oer number sn is defined by sn =
3(2n − 1) n−2 sn−1 − sn−2 , n+1 n+1
n > 2,
(37)
with s0 = 1 and s1 = 1; see [9, 17]. Doˇsli´c [4] , and Liu and Wang [14] proved the log-convexity of the sequence {sn }∞ n=0 . It is easy to see that r1 = 3. Let N0 = 3. We can check that (14) and (15) hold for N0 = 3. The following corollary follows from Theorem 2 and the fact ss23 > ss12 > ss10 . Corollary 7. The sequence {sn }∞ n=0 is strictly log-convex. Let Rn be the number of the set of all tree-like polyhexes with n + 1 hexagons. The sequence {Rn }∞ n=0 satisfies the recurrence Rn =
5(n − 2) 3(2n − 1) Rn−1 − Rn−2 , n+1 n+1
n > 2,
(38)
with R0 = 1 and R1 = 1; see [11]. The sequence {Rn }∞ n=0 is the sequence A002212 in Sloane’s database [16]. Liu and Wang [14] proved the log-convexity of the sequence {Rn }∞ n=0 . Let N0 = 3. Employing Theorem 2 and evaluating the values of R2 , R3 and R4 , we can prove the following corollary. the electronic journal of combinatorics 20(4) (2013), #P3
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Corollary 8. The sequence {Rn }∞ n=0 is strictly log-convex. Let wn be the number of walks on cubic lattice with n steps, starting and finishing on the x−y plane and never going below it. The sequence {wn }∞ n=0 has three-term recurrence relation wn =
12(n − 1) 4(2n + 1) wn−1 − wn−2 , n+2 n+2
n > 2,
(39)
with w0 = 1 and w1 = 4; see [10]. The sequence {wn }∞ n=0 is the sequence A005572 in Sloane’s database [16]. Liu and Wang [14] proved the log-convexity of the sequence {wn }∞ n=0 . Set N0 = 2. The following corollary follows from Theorem 2 and the fact wi+1 i > wwi−1 for i = 1, 2. wi Corollary 9. The sequence {wn }∞ n=0 is strictly log-convex. Let Fn be defined by Fn =
2n3 − 5n2 − n + 1 4n4 − n3 − n2 + 3n + 2 F − Fn−2 , n−1 n4 + 2n2 − 1 2n3 − 3n2 + 2n
n > 2,
(40)
with F0 = 1 and F1 = 1. By (13), we find r1 = 42. Set N0 = 42. It is easy to check i > FFi−1 for 3 6 i 6 42. that (14) and (15) hold for N0 = 42. We can also verify that FFi+1 i Hence, we can prove the following corollary. Corollary 10. The sequence {Fn }∞ n=2 is strictly log-convex. To conclude this paper, we remark that the method presented in this paper can be used to prove the log-convexity of some combinatorial sequences satisfied longer recurrence relations. The principle is the same.
Acknowledgements The authors would like to thank the anonymous referees for valuable suggestions and comments which resulted in a great improvement of the original manuscript.
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