By using the Newton interpolation formula, we generalize the recent identities on the Catalan triangle obtained by Miana and Romero as well as those of Chen and Chu. We further study divisibility properties of sums of products of binomial coefficients and an odd power of a natural number. For example, we prove that for all positive integers $n_1, ..., n_m$, $n_{m+1}=n_1$, and any nonnegative integer $r$, the expression $$n_1^{-1}{n_1+n_{m}\choose n_1}^{-1} \sum_{k=1}^{n_1}k^{2r+1}\prod_{i=1}^{m} {n_i+n_{i+1}\choose n_i+k}$$ is either an integer or a half-integer. Moreover, several related conjectures are proposed.
It is shown that Bernoulli numbers and tangent numbers (the derivatives of the tangent function at zero) can be obtained by means of easily defined triangles of numbers in several ways, some of them very similar to the Catalan triangle and a Motzkin-like triangle. Our starting point in order to show this is a new expression of Zeta(2n) involving Motzkin paths.
We prove two conjectures on sums of products of Catalan triangle numbers, which were originally conjectured by Miana, Ohtsuka, and Romero [Discrete Math. 340 (2017), 2388--2397]. The first one is proved by using Zeilberger's algorithm, and the second one is proved by establishing its $q$-analogue.
Let p be any odd prime. We mainly show that $$\sum_{k=1}^{p-1}binomial(3k,k)*2^k/k=0 (mod p)$$ and $$\sum_{k=1}^{p-1}2^{k-1}C_k^{(2)}=(-1)^{(p-1)/2}-1 (mod p),$$ where $C_k^{(2)}=binomial(3k,k)/(2k+1)$ is the $k$th Catalan number of order 2.
In this paper, we study congruences on sums of products of binomial coefficients that can be proved by using properties of the Jacobi polynomials. We give special attention to polynomial congruences containing Catalan numbers, second-order Catalan numbers, the sequence (\seqnum{A176898}) $S_n=\frac{{6n\choose 3n}{3n\choose 2n}}{2{2n\choose n}(2n+1)},$ and the binomial coefficients ${3n\choose n}$ and ${4n\choose 2n}$. As an application, we address several conjectures of Z.\ W.\ Sun on congruences of sums involving $S_n$ and we prove a cubic residuacity criterion in terms of sums of the binomia…
Let $p$ be an odd prime and let $a,m$ be integers with $a>0$ and $m \not\equiv0\pmod p$. In this paper we determine $\sum_{k=0}^{p^a-1}\binom{2k}{k+d}/m^k$ mod $p^2$ for $d=0,1$; for example, $$\sum_{k=0}^{p^a-1}\frac{\binom{2k}k}{m^k}\equiv\left(\frac{m^2-4m}{p^a}\right)+\left(\frac{m^2-4m}{p^{a-1}}\right)u_{p-(\frac{m^2-4m}{p})}\pmod{p^2},$$ where $(-)$ is the Jacobi symbol, and $\{u_n\}_{n\geqslant0}$ is the Lucas sequence given by $u_0=0$, $u_1=1$ and $u_{n+1}=(m-2)u_n-u_{n-1}$ for $n=1,2,3,\ldots$. As an application, we determine $\sum_{0<k<p^a,\, k\equiv r\pmod{p-1}}C_k$ modulo $p^2$ for…
In this paper we consider the generalized Catalan numbers F(s,n)= 1/((s-1)n+1) binom{sn}{n}. We find all $n$ such that for $p$ prime, p^q divides F(p^q,n), q>=1. As a byproduct we settle a question of Hough and the late Simion on the divisibility of the 4-Catalan numbers. We also prove that \binom{p^qn+1}{n}, p^q<=99999, is squarefree for n sufficiently large (explicit), and with the help of the generalized Catalan numbers we find the set of possible exceptions. As consequences, we obtain that binom{4n+1}{n}, binom{9n+1}{n} are squarefree for n> 2^{1518}, respectively n>3^{956}, with at most 2…
We present a new alternating convolution formula for the super Catalan numbers which arises as a generalization of two known binomial identities. We prove a generalization of this formula by using auxiliary sums, recurrence relations, and induction. By using a new method, we prove one interesting divisibility result with super Catalan numbers.
Let $\{U_n\}_{n\geq 0}$ be a Lucas sequence. Then the equation $$|U_n|=m_1!m_2!\cdots m_k!$$ with $1<m_1\leq m_2\leq \cdots\leq m_k$ implies $n\in \{1,2, 3, 4, 6, 8, 12\}$. Further the equation $$|U_n|=D_{m_1}D_{m_2}\cdots D_{m_k}, \qquad D_{m_i}\in \{B_{m_i}, C_{m_i}\}$$ with $1<m_1\leq m_2\leq \cdots\leq m_k$ implies $n\in \{1,2, 3, 4, 6, 8, 12, 16\}$. Here $B_m$ is the middle binomial coefficient $\binom{2m}{m}$ and $C_m$ is the Catalan number $\frac{1}{m+1}\binom{2m}{m}$.
The binomial interpolated transform of a sequence is a generalization of the well-known binomial transform. We examine a Pascal-like triangle, on which a binomial interpolated transform works between the left and right diagonals, focusing on binary recurrences. We give the sums of the elements in rows and in rising diagonals, further we define two special classes of these arithmetical triangles.
Let $p$ be a prime and let $a$ be a positive integer. In this paper we investigate $\sum_{k=0}^{p^a-1}\binom[(h+1)k,k+d]/m^k$ modulo a prime $p$, where $d$ and $m$ are integers with $-h<d<=p^a$ and $m\not=0 (mod p)$. We also study congruences involving higher-order Catalan numbers $C_k^{(h)}=\binom[(h+1)k,k]/(hk+1)$ and $\bar C_k^{(h)}=\binomal[(h+1)k,k]*h/(k+1)$. Our tools include linear recurrences and the theory of cubic residues. Here are some typical results in the paper. (i) If $p^a=1 (mod 6)$ then $$\sum_{k=1}^{p^a-1}\binom[3k,k]/6^k=2^{(p^a-1)/3}-1 (mod p).$$ Also, $$\sum_{k=0}^{p^a-1}…
In this paper we establish some new congruences involving central binomial coefficients as well as Catalan numbers. Let $p$ be a prime and let $a$ be any positive integer. We determine $\sum_{k=0}^{p^a-1}\binom{2k}{k+d}$ mod $p^2$ for $d=0,...,p^a$ and $\sum_{k=0}^{p^a-1}\binom{2k}{k+δ}$ mod $p^3$ for $δ=0,1$. We also show that $$C_n^{-1}\sum_{k=0}^{p^a-1}C_{p^an+k}=1-3(n+1)((p^a-1)/3) (mod p^2)$$ for every n=0,1,2,..., where $C_m$ is the Catalan number $\binom{2m}{m}/(m+1)$, and (-) is the Legendre symbol.
Let k and n be positive integers. We mainly show that $$(ln+1) | k\binom{kn+ln}{kn},$$ $$2\binom{kn}n | \binom {2n}{n}C_{2n}^{(k-1)}$$, $$\binom{kn}n | (2k-1)C_n\binom{2kn}{2n},$$ $$\binom{2n}n | (k+1)C_n^{(k-1)}\binom{2kn}{kn},$$ $$2^{k-1}\binom{2n}{n} | \binom{2(2^k-1)n}{(2^k-1)n}C_n^{(2^k-2)},$$ $$(6n+1)\binom{5n}{n} | \binom{3n-1}{n-1}C_{3n}^{(4)},$$ and $$\binom{3n}{n} | \binom{5n-1}{n-1}C_{5n}^{(2)},$$ where C_n denotes the Catalan number $\binom{2n}{n}/(n+1)$, and C_m^{(h)} refers to the Catalan number $\binom{(h+1)m}{m}/(hm+1)$ of order h.
In this paper we consider combinatorial numbers $C_{m, k}$ for $m\ge 1$ and $k\ge 0$ which unifies the entries of the Catalan triangles $ B_{n, k}$ and $ A_{n, k}$ for appropriate values of parameters $m$ and $k$, i.e., $B_{n, k}=C_{2n,n-k}$ and $A_{n, k}=C_{2n+1,n+1-k}$. In fact, some of these numbers are the well-known Catalan numbers $C_n$ that is $C_{2n,n-1}=C_{2n+1,n}=C_n$. We present new identities for recurrence relations, linear sums and alternating sum of $C_{m,k}$. After that, we check sums (and alternating sums) of squares and cubes of $C_{m,k}$ and, consequently, for $ B_{n, k}$ an…
The trinomial transform of a sequence is a generalization of the well-known binomial transform, replacing binomial coefficients with trinomial coefficients. We examine Pascal-like triangles under trinomial transform, focusing on the ternary linear recurrent sequences. We determine the sums and alternating sums of the elements in columns, and we give some examples of the trinomial transform triangle.
There exists an infinite series of ratios by which one can derive the Riemann zeta function $ζ(s)$ from Catalan numbers and central binomial coefficients which appear in the terms of the series. While admittedly the derivation is not deep it does indicate some combinatorial aspect to the Riemann zeta function. But we actually do find also four additional new closed formulas, which include a formula by which one can compute $ζ(s)$ for a countably infinite number of discrete positive values for $s$ where the formula contains Catalan numbers not in infinite series. The Riemann zeta function has a…
In this paper we initiate the study of products and sums divisible by central binomial coefficients. We show that 2(2n+1)binom(2n,n)| binom(6n,3n)binom(3n,n) for every n=1,2,3,... Also, for any nonnegative integers $k$ and $n$ we have $$\binom {2k}k | \binom{4n+2k+2}{2n+k+1}\binom{2n+k+1}{2k}\binom{2n-k+1}n$$ and $$\binom{2k}k | (2n+1)\binom{2n}nC_{n+k}\binom{n+k+1}{2k},$$ where $C_m$ denotes the Catalan number $\binom{2m}m/(m+1)=\binom{2m}m-\binom{2m}{m+1}$. Applying this result we obtain two sums divisible by central binomial coefficients.
We first prove that if $a$ has a prime factor not dividing $b$ then there are infinitely many positive integers $n$ such that $\binom {an+bn} {an}$ is not divisible by $bn+1$. This confirms a recent conjecture of Z.-W. Sun. Moreover, we provide some new divisibility properties of binomial coefficients: for example, we prove that $\binom {12n} {3n}$ and $\binom {12n} {4n}$ are divisible by $6n-1$, and that $\binom {330n} {88n}$ is divisible by $66n-1$, for all positive integers $n$. As we show, the latter results are in fact consequences of divisibility and positivity results for quotients of $…
The \emph{Hosoya triangle} is a triangular array where every entry is a product of two Fibonacci numbers. We use the geometry of this triangle to find new identities related to Fibonacci numbers. We give geometric interpretation for some well-known identities of Fibonacci numbers. For instance, the Cassini identity and the Catalan identity. We also extend some identities that hold in the Pascal triangle to the Hosoya triangle. For instance, the hockey stick extends from binomials to products of Fibonacci numbers and the rhombus property extends a binomial identity from the Pascal triangle to a…
For all positive integers n, we prove the following divisibility properties: $$(2n+3){2n\choose n} | 3{6n\choose 3n}{3n\choose n}, and (10n+3){3n\choose n} | 21{15n\choose 5n} {5n\choose n}.$$ This confirms two recent conjectures of Z.-W. Sun. Some similar divisibility properties are given. Moreover, we show that, for all positive integers m and n, the product $am{am+bm-1\choose am}{an+bn\choose an}$ is divisible by m+n. In fact, the latter result can be generalized to the q-binomial coefficients and q-integers case, which generalizes the positivity of q-Catalan numbers. We also propose severa…
We present a triangle of connections between the Sierpinski triangle, the sum-of-digits function, and the Binomial Theorem via a one-parameter family of Sierpinski matrices, which encodes a digital version of the Binomial Theorem.
In this note, by making use of a known hypergeometric series identity, I prove two Ramanujan-type series for the Catalan's constant. The convergence rate of these central binomial series surpasses those of all known similar series, including a classical formula by Ramanujan and a recent formula by Lupas. Interestingly, this suggests that an Apéry-like irrationality proof could be found for this constant.
We extend the digital binomial identity as given by Nguyen el al. to an identity in an arbitrary base $b$, by introducing the $b-$ary binomial coefficients. We then study the properties of these coefficients such as orthogonality, a link to Lucas' theorem and the corresponding $b-$ary Pascal triangles.
It is well known that for all $n\geq1$ the number $n+ 1$ is a divisor of the central binomial coefficient ${2n\choose n}$. Since the $n$th central binomial coefficient equals the number of lattice paths from $(0,0)$ to $(n,n)$ by unit steps north or east, a natural question is whether there is a way to partition these paths into sets of $n+ 1$ paths or $n+1$ equinumerous sets of paths. The Chung-Feller theorem gives an elegant answer to this question. We pose and deliver an answer to the analogous question for $2n-1$, another divisor of ${2n\choose n}$. We then show our main result follows fro…
The q-binomial coefficients are the polynomial cousins of the traditional binomial coefficients, and a number of identities for binomial coefficients can be translated into this polynomial setting. For instance, the familiar vanishing of the alternating sum across row n of Pascal's triangle is captured by the so-called Gaussian Formula. In this paper, we find a q-binomial congruence which synthesizes this result and Fleck's congruence for binomial coefficients.
We show that, for all positive integers $n_1, \ldots, n_m$, $n_{m+1}=n_1$, and any non-negative integers $j$ and $r$ with $j\leqslant m$, the expression $$ \frac{1}{[n_1]}{n_1+n_{m}\brack n_1}^{-1} \sum_{k=1}^{n_1}[2k][k]^{2r}q^{jk^2-(r+1)k}\prod_{i=1}^{m} {n_i+n_{i+1}\brack n_i+k} $$ is a Laurent polynomial in $q$ with integer cofficients, where $[n]=1+q+\cdots+q^{n-1}$ and ${n\brack k}=\prod_{i=1}^k(1-q^{n-i+1})/(1-q^i)$. This gives a $q$-analogue of a divisibility result on the Catalan triangle obtained by the first author and Zeng, and also confirms a conjecture of the first author and Zen…
The aim of this paper is to find a general formula to generate any row of Pascal's triangle as an extension of the concept of $\left(11\right)^{n}$. In this study, the visualization of each row of Pascal's triangle has been presented by extending the concept of the power of 11 to the power of 101, 1001, 10001, and so on. We briefly discuss how our proposed concept works for any $n$ by inserting an appropriate number of zeros between $1$ and $1$ (eleven), that is the concept of $\left(11\right)^{n}$ has been extended to $\left(1\Theta1\right)^{n}$, where $Θ$ represents the number of zeros. We h…
The first author recently introduced an integer sequence now numbered A355519 in OEIS. This sequence arose from counting bracket tournaments; its study evokes the analysis of the Catalan triangle (sequence A009766 in OEIS) and the related Catalan numbers, sequence A000108 in OEIS. We therefore introduce a general construction that places these sequences on the same footing and suggests many new integer sequences. We provide code for performing this construction and a lengthy list of examples. This construction determines a function from input sequences to output sequences. Some of the resultin…
For nonnegative integers $j$ and $n$ let $Θ(j,n)$ be the number of entries in the $n$-th row of Pascal's triangle that are not divisible by $2^{j+1}$. In this paper we prove that the family $j\mapstoΘ(j,n)$ usually follows a normal distribution. The method used for proving this theorem involves the computation of first and second moments of $Θ(j,n)$, and uses asymptotic analysis of multivariate generating functions by complex analytic methods, building on earlier work by Drmota (1994) and Drmota, Kauers and Spiegelhofer (2016).
In this paper, we introduce an analog of the Al-Karaji arithmetic triangle by substituting in the formula of the binomial coefficients the products by the least common multiples. Then, we give some properties and some open questions related to the obtained triangle.
Denote by $x$ a random infinite path in the graph of Pascal's triangle (left and right turns are selected independently with fixed probabilities) and by $d_n(x)$ the binomial coefficient at the $n$'th level along the path $x$. Then for a dense $G_δ$ set of $θ$ in the unit interval, $\{d_n(x)θ\}$ is almost surely dense but not uniformly distributed modulo 1.
The origin of this study is based on not only explicit formulas of finite sums involving higher powers of binomial coefficients, but also explicit evaluations of generating functions for this sums. It should be emphasized that this study contains both new results and literature surveys about some of the related results that have existed so far. With the aid of hypergeometric function, generating functions for a new family of the combinatorial numbers, related to finite sums of powers of binomial coefficients, are constructed. By using these generating functions, a number of new identities have…
The (generalised) Mellin transforms of certain Chebyshev and Gegenbauer functions based upon the Chebyshev and Gegenbauer polynomials, have polynomial factors $p_n(s)$, whose zeros lie all on the `critical line' $\Re\,s=1/2$ or on the real axis (called critical polynomials). The transforms are identified in terms of combinatorial sums related to H. W. Gould's S:4/3, S:4/2 and S:3/1 binomial coefficient forms. Their `critical polynomial' factors are then identified as variants of the S:4/1 form, and more compactly in terms of certain $_3F_2(1)$ hypergeometric functions. Furthermore, we extend t…
