arXiv:1602.00574v5 [math.CO] 19 Oct 2017 16,1C8 33F99. 11C08, 11B65, 146–171. (2018), 183 Theory Number J. Keywords 2010 upre yteNtoa aua cec onaino China of Foundation Science Natural National the by Supported and Then Schr¨oder number Abstract. of oevr o n d prime odd any for Moreover, where h supercongruence the sa plcto ecnr ojcue55i S4] nparticu in [S14a], in 5.5 Conjecture confirm we application an As ahmtc ujc Classification Subject D n ( D T ogune oyoil eanynme,Sh¨drnme,M Schr¨oder number, number, Delannoy Polynomial, Congruence, . x k RTMTCPOETE FDELANNOY OF PROPERTIES ARITHMETIC n and ) 1 sthe is (1) sthe is s UBR N SCHR AND NUMBERS n ( n 1 Define eateto ahmtc,NnigUniversity Nanjing Mathematics, of Department x s n = ) k X n k =0 D − ajn 103 epesRpbi fChina of Republic People’s 210093, Nanjing n 1 ( t eta rnma offiin and coefficient trinomial central -th 1 x n s n k X k X .W hwthat show We ). D n ( =0 − =1 n n x http://maths.nju.edu.cn/ nti ae eoti oesrrsn rtmtcproperties arithmetic surprising some obtain we paper this In . k 1 t eta eanynumber Delannoy central -th = ) ( T x n 1 k )  k p X M s k X − =0 n k k =0 n k +1 1  ( D  − ( p k k n k x 3) [email protected] ( −  ) n and x n 2 ∈ h-e Sun Zhi-Wei ) 1 − x s  Z k 1 k p x − ( +1 [ rmr 10,1B5 eodr 51,05A19, 05A10, Secondary 11B75; 11A07, Primary . x ai integer -adic x k k ( − ( 1) + x ∈ x 1 1 ) o all for 1)] + ( ) Z x ≡ n 1) + − (mod 0 DRNUMBERS ODER o all for ¨ k n − for x D k M n 6≡ n n ∼ k and , 1 = for p 0 0 = sthe is zwsun 2 n , ) − . 1 = , n , 2 (mod 1 1 , 1 = s , 3 k n , gat11571162). (grant 2 , . . . , 2 t ozi number. Motzkin -th 1 sthe is (1) . . . , , , 3 2 a epoethat prove we lar . . . . , , 3 p . . . . , ,w establish we ), tknnumber otzkin n t little -th 2 ZHI-WEI SUN

1. Introduction

For m, n N = 0, 1, 2,... , the ∈ { }

m n k Dm,n := 2 (1.1) N k k kX∈    in counts lattice paths from (0, 0) to (m, n) in which only east (1, 0), north (0, 1), and northeast (1, 1) steps are allowed (cf. R. P. Stanley [St99, p. 185]). The n-th central Delannoy number Dn = Dn,n has another well-known expression:

n n n + k n n + k 2k D = = . (1.2) n k k 2k k kX=0    kX=0    For n Z+ = 1, 2, 3,... , the n-th little Schr¨oder number is given by ∈ { } n k−1 sn := N(n,k)2 (1.3) kX=1 with the N(n,k) defined by

1 n n N(n,k) := Z. n k k 1 ∈   −  (See [Gr, pp.268–281] for certain combinatorial interpretations of the Narayana number N(n,k) = N(n,n +1 k).) For n N, the n-th large Schr¨oder number is given by − ∈ n n n + k 1 n n + k S := = C , (1.4) n k k k +1 2k k Xk=0    kX=0   where C denotes the 2k /(k +1) = 2k 2k . It is well known k k k − k+1 that S = 2s for every n = 1, 2, 3,... . Both little Schr¨oder numbers and large n n    Schr¨oder numbers have many combinatorial interpretations (cf. [St97] and [St99, pp. 178, 239-240]); for example, sn is the number of ways to insert parentheses into an expression of n + 1 terms with two or more items within a parenthesis, and Sn is the number of lattice paths from the point (0, 0) to (n,n) with steps (1, 0), (0, 1) and (1, 1) which never rise above the line y = x. Surprisingly, the central Delannoy numbers and Schr¨oder numbers arising nat- urally in enumerative combinatorics, have nice arithmetic properties. In 2011 the author [S11b] showed that

p−1 p−1 D S k ( 1)(p−1)/22E (mod p) and k 0 (mod p) k2 ≡ − p−3 6k ≡ Xk=1 Xk=1 PROPERTIES OF DELANNOY NUMBERS AND SCHRODER¨ NUMBERS 3 for any prime p > 3, where E0,E1,E2,... are the . In 2014 the author [S14a] proved that 1 n−1 (2k + 1)D2 Z for all n =1, 2, 3,..., n2 k ∈ Xk=0 and that p−1 2 D2 (mod p) for any odd prime p, k ≡ p k=0   · X where ( p ) denotes the Legendre symbol. Definition 1.1. We define n n 2 D (x) := xk(x + 1)n−k for n N, (1.5) n k ∈ Xk=0   and n s (x) := N(n,k)xk−1(x + 1)n−k for n Z+. (1.6) n ∈ k=1 X + Obviously Dn(1) = Dn for n N, and sn(1) = sn for n Z . In this paper we obtain somewhat∈ curious results involving∈ the polynomials Dn(x) and sn(x). Our first theorem is as follows. Theorem 1.1. (i) For any n Z+, we have ∈ 1 n−1 D (x)s (x) = W (x(x + 1)), (1.7) n k k+1 n Xk=0 where n W (x) = w(n,k)C xk−1 Z[x] (1.8) n k−1 ∈ kX=1 with 1 n 1 n + k w(n,k) = − Z. (1.9) k k 1 k 1 ∈  −  −  (ii) Let p be an odd prime. For any p-adic integer x, we have p−1

Dk(x)sk+1(x) Xk=0 p(1 x(x +1)) (mod p3) if x 0, 1 (mod p), − ≡ − ≡ 2p2 + 2x+1 p2(x2q (x) (x + 1)2q (x +1)) (mod p3) otherwise, ( x(x+1) p − p (1.10) p−1 where qp(z) denotes the Fermat quotient (z 1)/p for any p-adic integer z 0 (mod p). − 6≡

Remark 1.1. It is interesting to compare the new numbers w(n,k) with the Narayana numbers N(n,k). Clearly, Theorem 1.1 in the case x = 1 yields the following consequence. 4 ZHI-WEI SUN

Corollary 1.1. For any positive integer n, we have 1 n−1 n D s = w(n,k)C 2k−1 1 (mod2). (1.11) n k k+1 k−1 ≡ kX=0 Xk=1 Also, for any odd prime p we have p−1 D s 2p2(1 3q (2)) (mod p3). (1.12) k k+1 ≡ − p Xk=0

Remark 1.2. For the prime p = 588811, we have qp(2) 1/3 (mod p) and hence p−1 3 ≡ k=0 Dksk+1 0 (mod p ). In 2016 J.-C. Liu [L] confirmed the author’s conjec- ture (cf. [S11b,≡ Conjecture 1.1]) that P p−1 p−1 ( 1)k +3 D S 2p − (mod p4) for any prime p > 3. k k ≡ − k kX=1 kX=1 From Theorem 1.1 we can deduce a novel combinatorial identity. Corollary 1.2. For any n Z+, we have ∈ n n n + k C (n + 1)/2 n 2 k−1 = ⌊ ⌋ . (1.13) k k 1 ( 4)k−1 4n−1 n/2 Xk=1   −  − ⌊ ⌋

Remark 1.3. If we let an denote the left-hand side of (1.13), then the Zeilberger algorithm (cf. [PWZ, pp.101-119]) cannot find a closed form for an, and it only yields the following second-order : (n + 1)2a + (2n + 3)a (n + 1)(n + 3)a =0 for n =1, 2, 3,.... n n+1 − n+2 Now we give our second theorem which can be viewed as a supplement to The- orem 1.1. Theorem 1.2. Let p be any odd prime. Then p−1 kD (x)s (x) 2(x(x + 1))(p−1)/2 (mod p). (1.14) k k+1 ≡ kX=0 In particular, p−1 2 kD s 2 (mod p). (1.15) k k+1 ≡ p kX=0   In the next section we are going to show Theorems 1.1-1.2 and Corollary 1.2. In Section 3 we will give applications of Theorems 1.1-1.2 to central trinomial coefficients, Motzkin numbers, and their generalizations. Section 4 contains two related conjectures. Throughout this paper, for any polynomial P (x) and n N, we use [xn]P (x) to denote the coefficient of xn in P (x). ∈ PROPERTIES OF DELANNOY NUMBERS AND SCHRODER¨ NUMBERS 5

2. Proofs of Theorems 1.1-1.2 and Corollary 1.2

Recall the following definition given in [S12a] motivated by the large Schr¨oder numbers.

Definition 2.1. For n N we set ∈

n n n + k xk n n + k S (x) = = C xk. (2.1) n k k k +1 2k k kX=0    Xk=0   Lemma 2.1. We have

n n n + k D (x) = xk for n N. (2.2) n k k ∈ Xk=0    Also, for any n Z+ we have ∈ D (x) D (x)=2x(2n + 1)S (x) (2.3) n+1 − n−1 n and

(x + 1)sn(x) = Sn(x). (2.4)

Proof. For k,n N, we obviously have ∈

n n 2 k n 2 n j [xk]D (x) =[xk] xj(x + 1)n−j = − n j j k j j=0 j=0 X   X    −  n k n k n n + k = = k j k j k k j=0   X   −     with the help of the Chu-Vandermonde identity (cf. [G, (3.1)]). This proves (2.2). Now fix n Z+. For k N, by (2.2) we clearly have ∈ ∈ [xk+1](D (x) D (x)) n+1 − n−1 n +1+(k + 1) 2(k + 1) n 1+(k + 1) 2(k + 1) = − 2(k + 1) k +1 − 2(k + 1) k +1       2n +1 n + k 2k +2 2(2n + 1) n + k 2k = = 2k +1 2k k +1 k +1 2k k       k+1 =[x ]2x(2n + 1)Sn(x).

So (2.3) follows. 6 ZHI-WEI SUN

For each k N, it is apparent that ∈ n k k j−1 n−j [x ](x + 1)sn(x) =[x ](x + 1) N(n,j)x (x + 1) j=1 X n j k+1 n j = N(n,j) − + N(n,j) − k j k j +1 6 j=1 0

n n n + k x 1 k P (x) = − . n k k 2 kX=0      Lemma 2.2. Let n Z+. Then ∈ n n + k 2k 2k n(n + 1)S (x)2 = xk−1(x + 1)k+1 (2.5) n 2k k k +1 Xk=1     and

D (x) + D (x) n n + k 2k 2 2k +1 n−1 n+1 S (x) = xk(x + 1)k+1. (2.6) 2 n 2k k (k + 1)2 kX=0   

Remark 2.2. The identities (2.5) and (2.6) are (2.1) and (3.6) of the author’s paper [S12a] respectively. Lemma 2.3. For any m, n Z+ with m 6 n, we have the identity ∈ n k + m 2k +1 2m +1 m(m + 1) 2m − k(k + 1) kX=m     (2.7) (n m)(n + m + 1) n + m = − . n +1 2m   PROPERTIES OF DELANNOY NUMBERS AND SCHRODER¨ NUMBERS 7

Proof. When n = m, both sides of (2.7) vanish. Let m, n Z+ with n > m. If (2.7) holds, then ∈

n+1 k + m 2k +1 2m +1 m(m + 1) 2m − k(k + 1) kX=m     (n m)(n + m + 1) n + m = − n +1 2m   n + 1 + m 2(n +1)+1 + 2m +1 m(m + 1) 2m − (n + 1)(n + 2)     n + m +1 (n m)(n m + 1) m(m + 1)(2n + 3) = − − +2m +1 2m n +1 − (n + 1)(n + 2)     (n +1)+ m (n +1 m)((n +1)+ m + 1) = − . 2m (n +1)+1   In view of the above, we have proved Lemma 2.3 by induction.  Let A and B be integers. The Lucas sequence un(A,B)(n = 0, 1, 2,...) is defined by u0(A,B)=0, u1(A,B)= 1, and

u (A,B) = Au (A,B) Bu (A,B) for n =1, 2, 3,.... n+1 n − n−1 It is well known that if ∆ = A2 4B = 0 then − 6 αn βn u (A,B) = − for all n N, n α β ∈ − where α and β are the two roots of the quadratic equation x2 Ax + B = 0 (so that α + β = A and αβ = B). It is also known (see, e.g., [S10,− Lemma 2.3]) that for any odd prime p we have

∆ up(A,B) (mod p), and up−( ∆ )(A,B) 0 (mod p) if p ∤ B. ≡ p p ≡  

5 In particular, Fp ( ) (mod p) and p Fp−( 5 ) for any odd prime p, where Fn = ≡ p | p u (1, 1) with n N is the n-th . n − ∈ Lemma 2.4. Let p be an odd prime, and let x be any integer not divisible by p. Then 4x +1 4x +1 W (x) 1 (mod p). (2.8) p ≡ 2x p −    Moreover, if x 1/4 (mod p) then ≡ − W (x) 2p (mod p2), (2.9) p ≡ 8 ZHI-WEI SUN

otherwise we have

4x +1 4x +1 W (x) 2p + 1 xp−1 +(p + 1) 1 p ≡ 2x − p −     4x +1 4x +1 2 2 4x+1 2x + up−( 4x+1 ) 2x +1,x (mod p ). − 2−( p ) p p 4x     (2.10)

Proof. Clearly,

p−1 (p−1)/2 2p 1 p 1 1 − = 1 + 1 + p + 1 (mod p2). p 1 k ≡ k p k ≡  −  kY=1   kX=1  − 

(J. Wolstenholme [W] even showed that 2p−1 1 (mod p3) if p > 3.) Thus p−1 ≡ 

1 2p 2 2p 1 w(p, p) = = − 2(1 p) (mod p2), p p 1 p +1 p 1 ≡ −  −   − 

and

1 2p 2 1 2p 1 C = − = − (2p +1) (mod p2). p−1 p p 1 2p 1 p 1 ≡ −  −  −  − 

For each k =1,...,p 1, clearly −

1 p 1 p + k 1 p + k w(p, k) = − − k k 1 k 1 p +1  −  −  p 1 p j p + j = 1 + − k p +1 j · j   0

Therefore

p−1 p−1 k−1 Wp(x) =w(p, p)Cp−1x + w(p, k)Ck−1x kX=1 p−1 p 2(1 p)C xp−1 + 1 p + C ( x)k−1 ≡ − p−1 − k k−1 − k=1   X p (2 2p)C xp−1 +(p + 1) C ( x)k−1 +1 C ( x)p−1 ≡ − p−1 k−1 − − p−1 −  kX=2  p−1 1 + p 2 C ( x)k−1 k − k−1 − Xk=1   p−1 p +1 (1 3p)(1+2p)xp−1 +(p + 1) C ( x)k ≡ − − k − Xk=1 p−1 p 2k k ( x)k−1 − 2 k − k=1  X p−1 p−1 C p 2k 2p +1 xp−1 +(p + 1) k m k (mod p2), ≡ − mk − 2 kmk k=1 k=1  X X where m is an integer with m 1/x (mod p2). By [S10, Theorem 1.1], we have ≡ −

p−1 Ck p−1 m 4 ∆ 2 m 1 − 1 + up−( ∆ )(m 2, 1) (mod p ) mk ≡ − − 2 p − p − Xk=1    m 4 ∆ − 1 (mod p), ≡ − 2 p −   

where 1 1 4x +1 ∆ := m(m 4) 4 = (mod p2). − ≡ −x −x − x2   So (2.8) follows. If m 4 (mod p), then by [S12b, Lemma 3.5] we have 6≡

p−1 2k ( m)p−1 1 m 4 ∆ up−( ∆ )(2 m, 1) k − − + − p − kmk ≡ p 2 p p k=1  X   mp−1 1 m 4 ∆ up−( ∆ )(m 2, 1) = − − p − (mod p) p − 2 p p   10 ZHI-WEI SUN

and hence from the above we deduce that

W (x) 2p +1 xp−1 p ≡ − p−1 m 4 ∆ +(p + 1) m 1 − 1 + up−( ∆ )(m 2, 1) − − 2 p − p −     p mp−1 1 m 4 ∆ up−( ∆ )(m 2, 1) m − − p − − 2 p − 2 p p   ! m m 4 4x +1 2p +1 xp−1 + 1 (mp−1 1) (p + 1) − 1 ≡ − − 2 − − 2 p −      m 4 m 4x +1 − 1 up−( 4x+1 )(m 2, 1) − 2 − 2 p p −    2x +1 4x +1 4x +1 2p +1 xp−1 + (1 xp−1)+(p + 1) 1 ≡ − 2x − 2x p −    4x +1 1 4x +1 2 + 1 + up−( 4x+1 )(m 2, 1) (mod p ) 2x 2x p p −    which is equivalent to (2.10) since

( x)k−1u (m 2, 1) = u ( x(m 2), ( x)2) u (2x +1,x2) (mod p2) − k − k − − − ≡ k

for all k N. ∈ When m 4 (mod p) (i.e., x 1/4 (mod p)), we have ≡ ≡ −

p−1 2k p−1 2k k k 2q (2) (mod p) kmk ≡ k4k ≡ p k=1  k=1  X X by [ST, (1.12)], and hence

m 4 p W (x) 2p +1 xp−1 +(p + 1) mp−1 1 + − m2q (2) p ≡ − − 2 − 2 p   m 4 2p +1 xp−1 + mp−1 1 + − 4(2p−1 1) ≡ − − 2 − − 4x +1 2p +1 xp−1 +1 xp−1 2(22(p−1) 1) ≡ − − − 2x − − 2 p + (4x +1)+(1 xp−1)+(1 4p−1) ≡ − − 2 p + (4x +1)+1 (4x +1 1)p−1 ≡ − −  2(p + (4x +1)+(p 1)(4x + 1)) 2p (mod p2). ≡ − ≡ 

Therefore (2.9) is valid.  PROPERTIES OF DELANNOY NUMBERS AND SCHRODER¨ NUMBERS 11

Proof of Theorem 1.1. (i) Fix n Z+. For each k Z+, by (2.3) and Lemma 2.2 we have ∈ ∈ S (x) D (x) + D (x) S (x) x D (x) k = k−1 k+1 k (2k + 1) S (x)2 k−1 x +1 2 · x +1 − x +1 k k k + j 2j 2 2j +1 = (x(x + 1))j 2j j (j + 1)2 j=0 X    2k +1 k k + j 2j 2j (x(x + 1))j − k(k + 1) 2j j j +1 j=0 X     k k + j 2j 2 2j +1 2k +1 j = (x(x + 1))j 2j j (j + 1)2 − k(k + 1) · j +1 j=0 X      k k + j 2k +1 = C2 2j +1 j(j + 1) (x(x + 1))j. 2j j − k(k + 1) Xj=0     Combining this with (2.4) we obtain

k k + j 2k +1 D (x)s (x) = C2 2j +1 j(j + 1) (x(x + 1))j (2.11) k−1 k 2j j − k(k + 1) j=0 X     for any k Z+. Therefore, ∈ n n k k + j 2k +1 D (x)s (x) = C2 2j +1 j(j + 1) (x(x + 1))j k−1 k 2j j − k(k + 1) kX=1 kX=1 Xj=0     n n k + j 2k +1 =n + C2(x(x + 1))j 2j +1 j(j + 1) j 2j − k(k + 1) j=1 X Xk=j     n−1 (n j)(n + j + 1) n + j = C2(x(x + 1))j − j n +1 2j j=0 X   with the help of Lemma 2.3. It follows that

n−1 n−1 n n 1 n + j +1 D (x)s (x) = C (x(x+1))j − = nW (x(x+1)). k k+1 j j +1 j j n kX=0 Xj=0    For any k Z+, we have ∈ 1 n n + k 1 n 1 n + k w(n,k) = = − n k k 1 n +1 k 1 k   −   −   and hence w(n,k)=(n +1)w(n,k) nw(n,k) Z. So Wn(x) Z[x]. This proves part (i) of Theorem 1.1. − ∈ ∈ 12 ZHI-WEI SUN

(ii) Let x be any p-adic integer, and let y be an integer with y x(x + 1) (mod p2). By (1.7) we have ≡

p−1 D (x)s (x) = pW (x(x + 1)) pW (y) (mod p3). (2.12) k k+1 p ≡ p kX=0 If y 0 (mod p), then ≡ W (y) w(p, 1) + w(p, 2)y 1 x(x +1) (mod p2). p ≡ ≡ − 2 If x 1/2 (mod p) (i.e., y 1/4 (mod p)), then Wp(y) 2p (mod p ) by Lemma≡ − 2.4. Thus (1.10) holds≡ for −x 0, 1, 1/2 (mod p). ≡ Now assume that x 0, 1, 1/2≡ (mod− p),− i.e., y 0, 1/4 (mod p). Then 6≡ − − 6≡ − 4y +1 (2x + 1)2 = =1 p p     and u (2y +1,y2) u (x2 +(x + 1)2,x2(x + 1)2) p−1 ≡ p−1 ((x + 1)2)p−1 (x2)p−1 = − (x + 1)2 x2 − (x + 1)p−1 + xp−1 = (x + 1)p−1 xp−1 2x +1 − 2  (x + 1)p−1 xp−1 (mod p2). ≡2x +1 − Combining this with Lemma 2.4, we obtain  (2x + 1)2 W (y) 2p + 1 xp−1(x + 1)p−1 p ≡ 2x(x + 1) − (2x + 1)2 2  (2x(x +1)+1) (x + 1)p−1 xp−1 − 4x(x + 1) 2x +1 − (2x + 1)2  2p + 1 xp−1 +1 (x + 1)p−1 ≡ 2x(x + 1) − − 2x +1  (2x2 +2x + 1) (x + 1)p−1 xp−1 − 2x(x + 1) − 2x +1  =2p + x2(xp−1 1) (x + 1)2 (x + 1)p−1 1 (mod p2). x(x + 1) − − −  Therefore (1.10) holds in light of (2.12). So far we have completed the proof of Theorem 1.1.  Proof of Corollary 1.2. It is known (cf. [G, (3.133)]) that

1 ( 1)k/2 k /2k if 2 k, D = P (0) = − k/2 | k −2 k   ( 0  otherwise. PROPERTIES OF DELANNOY NUMBERS AND SCHRODER¨ NUMBERS 13

By (2.4) and [S11a, Lemma 4.3],

k/2 k 1 1 ( 1) Ck/2/2 if 2 k, sk+1 =2Sk+1 = − | −2 −2 0 otherwise.      Therefore

n−1 1 1 D s k −2 k+1 −2 Xk=0     2 ⌊(n−1)/2⌋ 2j 2 ( 1)k/2 k = − C = j . 2k k/2 k/2 (j + 1)16j 06k

By induction,

2j 2 m (2m + 1)2 2m 2 j = for all m N. (j + 1)16j (m + 1)16m m ∈ j=0  X   So we have

n−1 1 1 D s k −2 k+1 −2 kX=0     (2 (n 1)/2 + 1)2 2 (n 1)/2 2 (n + 1)/2 n 2 = ⌊ − ⌋ ⌊ − ⌋ = ⌊ ⌋ . (n + 1)/2 16⌊(n−1)/2⌋ (n 1)/2 4n−1 n/2 ⌊ ⌋  ⌊ − ⌋  ⌊ ⌋ On the other hand, by applying (1.7) with x = 1/2 we obtain − n−1 1 1 1 n n n + k C D s = nW = k−1 . k −2 k+1 −2 n −4 k k 1 ( 4)k−1 Xk=0       kX=1   −  − Combining these we get the desired identity (1.13). This concludes the proof.  Lemma 2.5. Let p be any odd prime. For each j =0,...,p, we have

2 2 (mod p) if j =(p 1)/2, C uj − (2.13) j ≡ 0 (mod p) otherwise,  where k + j 2k +1 u := (k 1) 2j +1 j(j + 1) . (2.14) j − 2j − k(k + 1) 6 j

Proof. Clearly, up = 0 and

p +(p 1) 2p +1 u =(p 1) − 2(p 1)+1 (p 1)p 0 (mod p). p−1 − 2(p 1) − − − p(p + 1) ≡  −    14 ZHI-WEI SUN

If (p 1)/2

2 p 1 C = − 2( 1)(p−1)/2 (mod p). (p−1)/2 p +1 (p 1)/2 ≡ −  −  As k +(p 1)/2 p +1 − 0 (mod p) forall k = ,... ,p, p 1 ≡ 2  −  we have

p k +(p 1)/2 p 1 p 1 p +1 2k +1 u (k 1) − 2 − +1 − (p−1)/2 ≡ − p 1 · 2 − 2 · 2 · k(k + 1) k=Xp−1  −    p 1 k +(p 1)/2 2k +1 (k 1) − ≡4 − p 1 k(k + 1) k=Xp−1  −  p 2 p 1+(p 1)/2 2(p 1)+1 p 1 p +(p 1)/2 2p +1 = − − − − + − − 4 (p 1)/2 (p 1)p 4 (p + 1)/2 p(p + 1)  −  −   p 2 2p 1 1

p+j f(p, j) 2j (j + 2)uj + 2(2j + 1)uj+1 = 2(j + 1)(j + 2)(2j + 3) for all j =0,...,p 1, where − f(p, j):=(p j)(p + j + 1) (2j + 3)2p2 (2j2 +8j + 7)p (j + 1)(j + 2) . − − − This implies that for 0 6 j < (p 3)/2 we have  − u 0 (mod p) = u 0 (mod p). j ≡ ⇒ j+1 ≡ Thus u 0 (mod p) for all j =0,..., (p 3)/2. j ≡ − PROPERTIES OF DELANNOY NUMBERS AND SCHRODER¨ NUMBERS 15

Combining the above, we immediately obtain the desired (2.13). 

Proof of Theorem 1.2. In view of (2.11),

p (k 1)D (x)s (x) − k−1 k Xk=1 p k k + j 2k +1 = (k 1) C2 2j +1 j(j + 1) (x(x + 1))j − 2j j − k(k + 1) Xk=1 Xj=0     p p p p(p 1) = (k 1) + (x(x + 1))jC2u = − + C2u (x(x + 1))j, − j j 2 j j j=1 j=1 Xk=1 X X

where uj is given by (2.14). Thus, by applying Lemma 2.5 we find that

p−1 1 kD (x)s (x) 2(x(x + 1))(p−1)/2 Z [x(x + 1)], (2.15) p k k+1 − ∈ p  kX=0 

where Zp denotes the ring of p-adic integers. Therefore (1.14) holds. (1.14) with x = 1 gives (1.15). This concludes the proof. 

3. Applications to central trinomial coefficients and Motzkin numbers

Let n N and b, c Z. The n-th generalized central trinomial coefficient ∈ ∈n 2 n n Tn(b, c) is defined to be [x ](x + bx + c) , the coefficient of x in the expansion of (x2 + bx + c)n. It is easy to see that

⌊n/2⌋ n 2k T (b, c) = bn−2kck. (3.1) n 2k k kX=0   

Note that Tn(1, 1) is the central trinomial coefficient Tn and Tn(2, 1) is the central 2n binomial coefficient n . Also, Tn(3, 2) coincides with the central Delannoy number D . Sun [S14a] also defined the generalized Motzkin number M (b, c) by n  n

⌊n/2⌋ n M (b, c) = C bn−2kck. (3.2) n 2k k kX=0  

Note that Mn(1, 1) is the usual Motzkin number Mn (whose combinatorial inter- pretations can be found in [St99, Ex.6.38]) and Mn(2, 1) is the Catalan number Cn+1. Also, Mn(3, 2) coincides with the little Schr¨oder number sn+1. The au- thor [S14a, S14b] deduced some congruences involving Tn(b, c) and Mn(b, c), and 16 ZHI-WEI SUN

proposed in [S14b] some conjectural series for 1/π involving Tn(b, c) such as

∞ 66k + 17 540√2 T (10, 112)3 = , (21133)k k 11π kX=0 ∞ 126k + 31 880√5 T (22, 212)3 = , ( 80)3k k 21π Xk=0 − ∞ 3990k + 1147 432 T (62, 952)3 = (195√14+94√2). ( 288)3k k 95π kX=0 −

Now we point out that Tn(b, c) and Mn(b, c) are actually related to the polyno- mials Dn(x) and sn+1(x). Lemma 3.1. Let b, c Z with d = b2 4c =0. For any n N we have ∈ − 6 ∈

n b/√d 1 Tn(b, c)=(√d) Dn − (3.3) 2 !

and

n b/√d 1 Mn(b, c)=(√d) sn+1 − . (3.4) 2 !

Proof. In view of (3.1) and (3.2),

⌊n/2⌋ T (b, c) n 2k b n−2k c k n = (√d)n 2k k √d d kX=0        and ⌊n/2⌋ n−2k k Mn(b, c) n b c = Ck . (√d)n 2k √d d kX=0       Note that b 2 c 4 =1. √ − d  d So, it suffices to show the polynomial identities

⌊n/2⌋ n 2k (2x + 1)n−2k(x(x + 1))k = D (x) (3.5) 2k k n Xk=0    and ⌊n/2⌋ n C (2x + 1)n−2k(x(x + 1))k = s (x). (3.6) 2k k n+1 Xk=0   PROPERTIES OF DELANNOY NUMBERS AND SCHRODER¨ NUMBERS 17

It is easy to verify (3.5) and (3.6) for n = 0, 1, 2. Let un(x) denote the left- hand side or the right-hand side of (3.5). By the Zeilberger algorithm (cf. [PWZ, pp. 101-119]), we have the recurrence

(n + 2)u (x)=(2x + 1)(2n + 3)u (x) (n + 1)u (x) for n =0, 1, 2,.... n+2 n+1 − n

Thus (3.5) is valid by induction. Let vn(x) denote the left-hand side or the right- hand side of (3.6). By the Zeilberger algorithm, we have the recurrence

(n + 4)v (x)=(2x + 1)(2n + 5)v (x) (n + 1)v (x) for n =0, 1, 2,.... n+2 n+1 − n So (3.6) also holds by induction. The proof of Lemma 3.1 is now complete.  With the help of Theorem 1.1, we are able to confirm Conjecture 5.5 of the author [S14a] by proving the following result. Theorem 3.1. Let b, c Z and d = b2 4c. ∈ − (i) For any n Z+, we have ∈

1 n−1 n T (b, c)M (b, c)dn−1−k = w(n,k)C ck−1dn−k Z. (3.7) n k k k−1 ∈ Xk=0 kX=1 Moreover, for any odd prime p not dividing cd, we have

p−1 T (b, c)M (b, c) pb2 d k k 1 (mod p2), (3.8) dk ≡ 2c p − Xk=0    and furthermore

p−1 Tk(b, c)Mk(b, c) dk kX=0 pb2 d p2 d 1 + b2 q (d) q (c) + d (3.9) ≡ 2c p − 2c p − p p −         2 pb d 2 2 3 d 2c + d up−( d )(b 2c, c ) (mod p ). − 2−( p ) p p − 4c    (ii) For any odd prime p not dividing d, we have

p−1 kT (b, c)M (b, c) cd k k 2 (mod p). (3.10) dk ≡ p kX=0  

Proof. (i) Let’s first prove (3.7) for any n Z+. ∈ 18 ZHI-WEI SUN

We first consider the case d = 0, i.e., c = b2/4. In this case, for any k N we have ∈ b k b k 2k T (b, c) = T (2, 1) = k 2 k 2 k       and b k b k M (b, c) = M (2, 1) = C . k 2 k 2 k+1     Thus

n−1 1 T (b, c)M (b, c)dn−1−k n k k kX=0 T (b, c)M (b, c) 1 b2 n−1 2(n 1) = n−1 n−1 = − C n n 4 n 1 n     n−1 n−1 − =c Cn−1Cn = w(n,n)Cn−1c

and hence (3.7) is valid. Now assume that d = 0. By Lemma 3.1 and (1.7), we have 6 n−1 1 Tk(b, c)Mk(b, c) n dk Xk=0 1 n−1 b/√d 1 b/√d 1 = Dk − sk+1 − n 2 ! 2 ! Xk=0 b/√d 1 b/√d +1 b2/d 1 c =Wn − = Wn − = Wn 2 · 2 ! 4 d     and hence (3.7) holds in view of (1.8). Below we suppose that p is an odd prime not dividing cd. From the above, we have p−1 T (b, c)M (b, c) c k k = pW pW (x) (mod p3), (3.11) dk p d ≡ p Xk=0   where x is an integer with x c/d (mod p2). As p ∤ d and d(4x +1) 4c + d = b2 (mod p2), we have ≡ ≡

4x +1 d2(4x + 1) b2d = = . p p p       In view of Lemma 2.4,

4c/d +1 4x +1 b2 d W (x) 1 1 (mod p). p ≡ 2c/d p − ≡ 2c p −       PROPERTIES OF DELANNOY NUMBERS AND SCHRODER¨ NUMBERS 19

Combining this with (3.11) we immediately obtain (3.8). Now we show (3.9). If x 1/4 (mod p) (i.e., p b), then by (2.9) we have ≡ − | p W (x) 2p (4c b2) (mod p2) p ≡ ≡ 2c − and hence (3.9) holds by (3.11). Below we assume p ∤ b. Then (2x + 1)2 4x2 4x +1 b2d d − = = = , p p p p         2 p up−( d )(2x +1,x ) and | p p−1−( d ) 2 d p u d 2x +1,x p−( p ) 2 2 =u d (d(2x + 1),d x )  p−( p ) 2 2 2 2 up−( d )(2c + d, c ) = up−( d )(b 2c, c ) (mod p ). ≡ p p − So, applying (2.10) we get 4c/d +1 c p−1 d W (x) 2p + 1 +(p + 1) 1 p ≡ 2c/d − d p −       4c/d +1 c d d ( p ) 2 2 d 2 + d up−( d )(b 2c, c ) − 2−( p ) d p p − 4(c/d)    b2 d 2p + dp−1 1 (cp−1 1)+(p + 1) 1 ≡ 2c − − − p −     2 b d 2 2 2 d 2c + d up−( d )(b 2c, c ) (mod p ). − 2−( p ) p p − 4c    This, together with (3.11), yields the desired (3.9). (ii) Fix an odd prime p not dividing d. Let x = b/√d 1. Then − b/√d 1 b/√d +1 b2/d 1 c x(x +1) = − = − = 2 · 2 4 d is a p-adic integer. Thus, with the help of (2.15), we have

p−1 c (p−1)/2 cd kD (x)s (x) 2 2 (mod p). k k+1 ≡ d ≡ p kX=0     Combining this with Lemma 3.1, we immediately obtain (3.10). In view of the above, we have proved Theorem 3.1.  Let ω denote the primitive cubic root ( 1 + √ 3)/2 of unity. Then ω +¯ω = 1 and ωω¯ = 1. So, − − − ωn ω¯n ( 3ω)n ( 3¯ω)n u ( 1, 1) = − = 0 and u (3, 9) = − − − =0 n − ω ω¯ n ( 3ω) ( 3¯ω) − − − − for any n N with 3 n. In view of this, Theorem 3.1 in the cases b = c 1, 3 yields the∈ following consequence.| ∈ { } 20 ZHI-WEI SUN

Corollary 3.1. For any positive integer n, we have

1 n−1 n T M ( 3)n−1−k = w(n,k)C ( 3)n−k Z (3.12) n k k − k−1 − ∈ Xk=0 kX=1 and 1 n−1 T (3, 3)M (3, 3) n k k = ( 1)k−1w(n,k)C Z. (3.13) n ( 3)k − k−1 ∈ kX=0 − kX=1 Moreover, for any prime p > 3 we have

p−1 T M p p p2 p k k 1 + q (3) + +3 (mod p3), (3.14) ( 3)k ≡ 2 3 − 2 p 3 Xk=0 −       

p−1 T (3, 3)M (3, 3) 3p p p2 p k k 1 + 3 +1 (mod p3), (3.15) ( 3)k ≡ 2 3 − 2 3 Xk=0 −        p−1 p−1 kT M p kT (3, 3)M (3, 3) 1 k k 2 (mod p) and k k 2 − (mod p). ( 3)k ≡ 3 ( 3)k ≡ p k=0 −   k=0 −   X X (3.16)

Remark 3.1. Let p > 3 be a prime. In the case p 2 (mod 3), the author (cf. [S14a, Conjecture 5.6]) even conjectured that ≡

p−1 T (3, 3)M (3, 3) k k p3 p2 3p (mod p4) ( 3)k ≡ − − Xk=0 − which is stronger than (3.15). The author’s conjectural supercongruences (cf. [S14a, Conjecture 1.1(ii)])

p−1 p−1 p p M 2 (2 6p) (mod p2), kM 2 (9p 1) (mod p2), k ≡ − 3 k ≡ − 3 Xk=0   Xk=0   and p−1 4 p p p T M + 1 9 (mod p2) k k ≡ 3 3 6 − 3 Xk=0      remain open. We also observe that

p−1 1 5 p kT M − (mod p). k k ≡ p − 3 3 kX=0     PROPERTIES OF DELANNOY NUMBERS AND SCHRODER¨ NUMBERS 21

The Lucas numbers L0,L1,L2,... are given by

L0 =2, L1 =1, and Ln+1 = Ln + Ln−1 for n =1, 2, 3,....

+ It is easy to see that Ln =2Fn+1 Fn =2Fn−1 + Fn for all n Z . Thus, for any odd prime p = 5 we have − ∈ 6 p p L ( p ) =2Fp F ( p ) 2 (mod p) p− 5 − 5 p− 5 ≡ 5     and hence

p p 2 p−( 5 ) 2 p−( 5 ) (α ) (β ) p 2 u ( p )(3, 1) = − = F ( p )L ( p ) 2 F ( p ) (mod p ), p− 5 α2 β2 p− 5 p− 5 ≡ 5 p− 5 −   where α = (1+ √5)/2 and β = (1 √5)/2. Note also that − u (3 5, 52)=5n−1u (3, 1)=5n−1F L for any n N. n × n n n ∈ Thus Theorem 3.1 with (b, c)=(1, 1), (5, 5) leads to the following corollary. − Corollary 3.2. For any n Z+, we have ∈ 1 n−1 n T (1, 1)M (1, 1)5n−1−k = ( 1)k−1w(n,k)C 5n−k Z (3.17) n k − k − − k−1 ∈ Xk=0 kX=1 and n−1 n 1 T (5, 5)M (5, 5) k k = w(n,k)C Z. (3.18) n 5k k−1 ∈ Xk=0 Xk=1 Also, for any prime p =2, 5 we have the congruences 6 p−1 2 Tk(1, 1)Mk(1, 1) p p p p − k − 1 + 5 qp(5) 5 ≡ 2 − 5 2 − 5 − (3.19) Xk=0        p p 3 + 5 2 F ( p ) (mod p ), 2 − 5 p− 5    p−1 T (5, 5)M (5, 5) 5p p p2 p k k 1 + 5 1 5k ≡ 2 5 − 2 5 − k=0 (3.20) X        5p p 3 1+2 F ( p ) (mod p ), − 2 5 p− 5 and   

p−1 p−1 5 kT (1, 1)M (1, 1) kT (5, 5)M (5, 5) − k − k − k k 2 (mod p). (3.21) p 5k ≡ 5k ≡   Xk=0 Xk=0 22 ZHI-WEI SUN

4. Two related conjectures

In view of (1.8) and (1.9), we can easily see that

2 W1(x)=1, W2(x)=2x +1, W3(x)=10x +5x +1, 3 2 W4(x)=70x + 42x +9x +1.

Applying the Zeilberger algorithm (cf. [PWZ, pp.101-119]) via Mathematica 9, we obtain the following third-order recurrence with n Z+: ∈ 2 (n + 3) (n + 4)(2n + 3)Wn+3(x) 2 2 =(n + 3)(2n + 5)(4x(2n + 3) +3n + 11n + 10)Wn+2(x) (4.1) (n + 1)(2n + 3)(4x(2n + 5)2 +3n2 + 13n + 14)W (x) − n+1 2 + n(n + 1) (2n + 5)Wn(x).

(This is a verified result, not a conjecture.) For any n Z+, we clearly have w(n,n) = C . For the polynomial ∈ n n k−1 wn(x) := w(n,k)x , (4.2) Xk=1 we have the relation w ( 1 x)=( 1)n−1w (x) (4.3) n − − − n since

n k 1 ( 1)n−k − w(n,k) = w(n, m) forall m =1,... ,n, (4.4) − m 1 kX=m  −  which can be deduced with the help of the Chu-Vandermonde identity in the fol- lowing way:

n k 1 ( 1)n−k − w(n,k) − m 1 kX=m  −  n 1 n ( 1)n−k n m n 2 = − − − − − ( 1)k−1 m 1 k n k k 1 −  −  kX=m  −  −  ( 1)n n 1 n n m n 1 = − − − − − n +1 m 1 n k k  −  kX=m  −   ( 1)n n 1 m 1 = − − − − = w(n, m). n +1 m 1 n  −   Via the Zeilberger algorithm we obtain the recurrence

(n + 3)w (x)=(2x + 1)(2n + 3)w (x) nw (x) for n =1, 2, 3,... (4.5) n+2 n+1 − n PROPERTIES OF DELANNOY NUMBERS AND SCHRODER¨ NUMBERS 23

As w2(x)=2x+1, this recurrence implies that w2n( 1/2) = 0 and hence w2n(x)/(2x+ 1) Z[x] for all n Z+. We also note that − ∈ ∈ ∞ 1 y 2xy (y 1)2 4xy w (x)yn = − − − − − , (4.6) n 2x(x + 1)y n=1 p X while ∞ 1 y y2 6y +1 S yn = − − − . n 2y n=0 p X Now we pose two conjectures for further research. Conjecture 4.1. For any integer n > 1, all the polynomials w (x) w (x), 2n and W (x) 2n−1 2x +1 n are irreducible over the field of rational numbers. Conjecture 4.2. (i) For any n Z+, we have ∈ 1 n−1 f (x) := D (x)R (x) Z[x], (4.7) n n k k ∈ kX=0 where k k k k + l xl k + l 2l xl R (x) := = . (4.8) k l l 2l 1 2l l 2l 1 Xl=0    − Xl=0    − Also, f2(x),f3(x),... are all irreducible over the field of rational numbers, and n−1 1 f (1) = D R ( 1)n (mod 32) n n k k ≡ − k=0 + X for each n Z , where Rk = Rk(1). (ii) Let p∈be any odd prime. Then p−1 2 3 4 p +8p qp(2) 2p Ep−3 (mod p ) if p 1 (mod 4), DkRk − − ≡ (4.9) ≡ 5p (mod p3) if p 3 (mod 4). kX=0  − ≡ Also, p−1 D R 1 k k 4 − q (2) (mod p), (4.10) k ≡ − p p Xk=1    p−1 1 3 1 kD R + p 1 2 − (mod p2), (4.11) k k ≡ 2 2 − p Xk=1    and p−1 xp−1 kD (x)R (x) (mod p). k k ≡ 2 kX=1 Acknowledgment. The author would like to thank the referee for helpful com- ments. 24 ZHI-WEI SUN

References

[G] H. W. Gould, Combinatorial Identities, Morgantown Printing and Binding Co., 1972. [Gr] R. P. Grimaldi, Fibonacci Numbers and Catalan Numbers: An Introduction, John Wiley & Sons, New Jersey, 2012. [L] J.-C. Liu, A supercongruence involving Delannoy numbers and Schr¨oder numbers, J. 168 (2016), 117–127. [PWZ] M. Petkovˇsek, H. S. Wilf and D. Zeilberger, A = B, A K Peters, Wellesley, 1996. [St97] R. P. Stanley, Hipparchus, Plutarch, Schr¨oder, and Hough, Amer. Math. Monthly 104 (1997), 344–350. [St99] R. P. Stanley, Enumerative Combinatorics, Vol. 2, Cambridge Univ. Press, Cambridge, 1999. [S10] Z.-W. Sun, Binomial coefficients, Catalan numbers and Lucas quotients, Sci. China Math. 53 (2010), 2473–2488. [S11a] Z.-W. Sun, On congruences related to central binomial coefficients, J. Number Theory 131 (2011), 2219–2238. [S11b] Z.-W. Sun, On Delannoy numbers and Schr¨oder numbers, J. Number Theory 131 (2011), 2387–2397. [S12a] Z.-W. Sun, On sums involving products of three binomial coefficients, Acta Arith. 156 (2012), 123–141. [S12b] Z.-W. Sun, On sums of binomial coefficients modulo p2, Colloq. Math. 127 (2012), 39-54. [S14a] Z.-W. Sun, Congruences involving generalized central trinomial coefficients, Sci. China Math. 57 (2014), 1375–1400. [S14b] Z.-W. Sun, On sums related to central binomial and trinomial coefficients, in: M. B. Nathanson (ed.), Combinatorial and Additive Number Theory: CANT 2011 and 2012, Springer Proc. in Math. & Stat., Vol. 101, Springer, New York, 2014, pp. 257-312. [ST] Z.-W. Sun and R. Tauraso, New congruences for central binomial coefficients, Adv. in Appl. Math. 45 (2010), 125–148. [W] J. Wolstenholme, On certain properties of prime numbers, Quart. J. Appl. Math. 5 (1862), 35–39.