Several identities in the Catalan triangle |
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| Authors: | Zhizheng Zhang Bijun Pang |
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| Affiliation: | (1) Department of Mathematics, Nanjing University, Nanjing, 210093, People’s Republic of China;(2) State Key Laboratory of Novel Software Technology, Nanjing University, Nanjing, 210093, People’s Republic of China |
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| Abstract: | In this paper, we first establish several identities for the alternating sums in the Catalan triangle whose (n, p) entry is defined by B n, p = $
tfrac{p}
{n}left( {_{n - p}^{2n} } right)
$
tfrac{p}
{n}left( {_{n - p}^{2n} } right)
. Second, we show that the Catalan triangle matrix C can be factorized by C = FY = ZF, where F is the Fibonacci matrix. From these formulas, some interesting identities involving B n, p and the Fibonacci numbers F n are given. As special cases, some new relationships between the well-known Catalan numbers C n and the Fibonacci numbers are obtained, for example: | $
C_n = F_{n + 1} + sumlimits_{k = 3}^n {left{ {1 - frac{{(k + 1)(k5 - 6)}}
{{4(2k - 1)(2k - 3)}}} right}C_k F_{n - k + 1} } ,
$
C_n = F_{n + 1} + sumlimits_{k = 3}^n {left{ {1 - frac{{(k + 1)(k5 - 6)}}
{{4(2k - 1)(2k - 3)}}} right}C_k F_{n - k + 1} } ,
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| Keywords: | |
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