关于中心三项式系数的超同余式

Congruences involving generalized central trinomial coefficients

For integers $b$ and $c$ the generalized central trinomial coefficient $T_n(b,c)$ denotes the coefficient of $x^n$ in the expansion of $(x^2+bx+c)^n$. Those $T_n=T_n(1,1)\ (n=0,1,2,\ldots)$ are the usual central trinomial coefficients, and $T_n(3,2)$ coincides with the Delannoy number $D_n=\sum_{k=0}^n\binom nk\binom{n+k}k$ in combinatorics. We investigate congruences involving generalized central trinomial coefficients systematically. Here are some typical results: For each $n=1,2,3,\ldots$ we have $$\sum_{k=0}^{n-1}(2k+1)T_k(b,c)^2(b^2-4c)^{n-1-k}\equiv0\pmod{n^2}$$ and in particular $n^2\mi…

Some new results about $q$-trinomial coefficients

In this paper, we present several new congruences on the $q$-trinomial coefficients introduced by Andrews and Baxter. A new congruence on sums of central $q$-binomial coefficients is also established.

On Motzkin numbers and central trinomial coefficients

The Motzkin numbers $M_n=\sum_{k=0}^n\binom n{2k}\binom{2k}k/(k+1)$ $(n=0,1,2,\ldots)$ and the central trinomial coefficients $T_n$ ($n=0,1,2,\ldots)$ given by the constant term of $(1+x+x^{-1})^n$, have many combinatorial interpretations. In this paper we establish the following surprising arithmetic properties of them with $n$ any positive integer: $$\frac2n\sum_{k=1}^n(2k+1)M_k^2\in\mathbb Z,$$ $$\frac{n^2(n^2-1)}6\,\bigg|\,\sum_{k=0}^{n-1}k(k+1)(8k+9)T_kT_{k+1},$$ and also $$\sum_{k=0}^{n-1}(k+1)(k+2)(2k+3)M_k^23^{n-1-k}=n(n+1)(n+2)M_nM_{n-1}.$$

On sums related to central binomial and trinomial coefficients

A generalized central trinomial coefficient $T_n(b,c)$ is the coefficient of $x^n$ in the expansion of $(x^2+bx+c)^n$ with $b,c\in\mathbb Z$. In this paper we investigate congruences and series for sums of terms related to central binomial coefficients and generalized central trinomial coefficients. The paper contains many conjectures on congruences related to representations of primes by certain binary quadratic forms, and 62 proposed new series for $1/π$ motivated by congruences and related dualities.

Tetrahedron trinomial coefficient transform

We introduce the tetrahedron trinomial coefficient transform which takes a Pascal-like arithmetical triangle to a sequence. We define a Pascal-like infinite tetrahedron H, and prove that the application of the tetrahedron trinomial transform to one face T of H provides the opposite edge E to T in H. It follows from the construction that the other directions in H parallel to E can be obtained similarly. In case of Pascal's triangle the sequence generated by the trinomial transform coincides the binomial transform of the central binomial coefficients.

Supercongruences related to ${}_3F_2(1)$ involving harmonic numbers

We show various supercongruences for truncated series which involve central binomial coefficients and harmonic numbers. The corresponding infinite series are also evaluated.

广义中心三项式系数的参数化 Catalan 型同余

A parametric Catalan-type congruence for generalized central trinomial coefficients

本文针对广义中心三项式系数，给出一个参数化 Catalan 型同余式，解决 Wang 和 Cui 的求和问题，并覆盖 $p$ 整除 $b$ 或 $b-2c$ 的情形。

Asymptotics of generalised trinomial coefficients

It is shown how to obtain an asymptotic expansion of the generalised central trinomial coefficient $[x^n](x^2 + bx + c)^n$ by means of singularity analysis, thus proving a conjecture of Zhi-Wei Sun.

涉及广义中心三项式系数四次方的一些同余式

Some Congruences Involving Fourth Powers of Generalized Central Trinomial Coefficients

本文研究了广义中心三项式系数四次方的同余性质，针对特定情形证明了相关求和公式在模p³下的表达式。

Some congruences involving binomial coefficients

Binomial coefficients and central trinomial coefficients play important roles in combinatorics. Let $p>3$ be a prime. We show that $$T_{p-1}\equiv\left(\frac p3\right)3^{p-1}\ \pmod{p^2},$$ where the central trinomial coefficient $T_n$ is the constant term in the expansion of $(1+x+x^{-1})^n$. We also prove three congruences modulo $p^3$ conjectured by Sun, one of which is $$\sum_{k=0}^{p-1}\binom{p-1}k\binom{2k}k((-1)^k-(-3)^{-k})\equiv \left(\frac p3\right)(3^{p-1}-1)\ \pmod{p^3}.$$ In addition, we get some new combinatorial identities.

关于广义中心三项式系数和 Legendre 多项式的新整除性质

On new divisibility properties of generalized central trinomial coefficients and Legendre polynomials

本文提出了 $m=2$ 时整除和式 $B(n,m,a,b)$ 的 $a+b$ 最高幂次新公式，并据此确定了中心 Dellanoy 数等的完整 3-adic 赋值及 Legendre 多项式的整除性质。

On a conjecture of Deutsch, Sagan, and Wilson

We prove a recent conjecture due to Deutsch, Sagan, and Wilson stating that the finite sequence obtained from the first p central trinomial coefficients modulo p by replacing nonzero terms by 1's is palindromic, for any prime number p > 3. Addendum: the result was proved before almost in the same way by Tony D. Noe: On the Divisibility of Generalized Central Trinomial Coefficients, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7 http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Noe/noe35.html

Arithmetic properties of Delannoy numbers and Schröder numbers

Define $$D_n(x)=\sum_{k=0}^n\binom nk^2x^k(x+1)^{n-k}\ \ \ \mbox{for}\ n=0,1,2,\ldots$$ and $$s_n(x)=\sum_{k=1}^n\frac1n\binom nk\binom n{k-1}x^{k-1}(x+1)^{n-k}\ \ \ \mbox{for}\ n=1,2,3,\ldots.$$ Then $D_n(1)$ is the $n$-th central Delannoy number $D_n$, and $s_n(1)$ is the $n$-th little Schröder number $s_n$. In this paper we obtain some surprising arithmetic properties of $D_n(x)$ and $s_n(x)$. We show that $$\frac1n\sum_{k=0}^{n-1}D_k(x)s_{k+1}(x)\in\mathbb Z[x(x+1)]\ \quad\mbox{for all}\ n=1,2,3,\ldots.$$ Moreover, for any odd prime $p$ and $p$-adic integer $x\not\equiv0,-1\pmod p$, we est…

涉及p进对偶序列的新同余式

New Congruences Involving $p$-adic dual sequences

本文研究涉及中心二项式系数与p进对偶序列的模$p^2$同余式，证明相关同余关系，建立新公式并证明Z.-W. Sun关于广义中心三项式系数的部分猜想。