# Catalan numbers

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## 4 matching pages

##### 1: 26.5 Lattice Paths: Catalan Numbers
$C\left(n\right)$ is the Catalan number. …
26.5.3 $C\left(n+1\right)=\sum_{k=0}^{n}C\left(k\right)C\left(n-k\right),$
##### 2: 26.6 Other Lattice Path Numbers
###### §26.6(iv) Identities
26.6.13 $M(n)=\sum_{k=0}^{n}(-1)^{k}\genfrac{(}{)}{0.0pt}{}{n}{k}C\left(n+1-k\right),$
26.6.14 $C\left(n\right)=\sum_{k=0}^{2n}(-1)^{k}\genfrac{(}{)}{0.0pt}{}{2n}{k}M(2n-k).$
##### 3: 26.1 Special Notation
 $\genfrac{(}{)}{0.0pt}{}{m}{n}$ binomial coefficient. … Catalan number. …
##### 4: 25.11 Hurwitz Zeta Function
25.11.22 $\zeta'\left(1-2n,\tfrac{1}{2}\right)=-\frac{B_{2n}\ln 2}{n\cdot 4^{n}}-\frac{(% 2^{2n-1}-1)\zeta'\left(1-2n\right)}{2^{2n-1}},$ $n=1,2,3,\dots$.
25.11.33 $h(n)=\sum_{k=1}^{n}k^{-1}.$
25.11.39 $\sum_{k=2}^{\infty}\frac{k}{2^{k}}\zeta\left(k+1,\tfrac{3}{4}\right)=8G,$
where $G$ is Catalan’s constant:
25.11.40 $G\equiv\sum_{n=0}^{\infty}\frac{(-1)^{n}}{(2n+1)^{2}}=0.91596\;55941\;772\dots.$