论文标题
关于卢卡斯序列的成员，该序列是阶乘的产物或中间二项式系数和加泰罗尼亚数字的产物
On members of Lucas sequences which are either products of factorials or product of middle binomial coefficients and Catalan numbers
论文作者
论文摘要
令$ \ {u_n \} _ {n \ geq 0} $为lucas序列。然后公式$$ | u_n | = m_1！m_2！\ cdots m_k！$$，$ 1 <m_1 \ leq m_2 \ leq \ leq \ cdots \ cdots \ leq m_k $ in \ in \ in \ in \ {1,2、3、4、6、6、8、12 \} $。再加上公式$$ | u_n | = d_ {m_1} d_ {m_2} \ cdots d_ {m_k}，\ qquad d_ {m_i} \ in \ in \ in \ {b_i {m_i {m_i}，c_i {m_i} m_k $表示$ n \ in \ {1,2、3、4、6、8、12、16 \} $。这里$ b_m $是中间二项式系数$ \ binom {2m} {m} {m} $，$ c_m $是加泰罗尼亚号$ \ frac {1} {1} {m+1} \ binom {2m {2m} {m} {m} {m} $。
Let $\{U_n\}_{n\geq 0}$ be a Lucas sequence. Then the equation $$|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 $\frac{1}{m+1}\binom{2m}{m}$.
