@MISC{CATALAN2.MIZ,
  AUTHOR = {P\c{a}k, Karol},
  TITLE = {The {C}atalan Numbers. {P}art {II}},
  NOTE = {This work has been partially supported by the KBN grant 4 T11C 039 24 and the FP6 IST grant TYPES No. 510996.},
  SECTION1 = {Preliminaries},
  SECTION2 = {The Recurrence Relation for the Catalan Numbers},
  SECTION3 = {Cauchy Product},
  SECTION4 = {The Generating Function for the Catalan Numbers},
  DAY = {31},
  MONTH = {October},
  YEAR = {2006},
  ADDRESS1 = {Institute of Mathematics\\University of Bia{\l}ystok\\Akademicka 2, 15-267 Bia{\l}ystok, Poland},
  SUMMARY = {In this paper, we define sequence dominated by $0$, 
  in which every initial fragment contains more zeroes than ones. 
  If $n \geq 2 \cdot m$ and $n>0$, then the number of sequences 
  dominated by $0$ the length $n$ including $m$ of ones, 
  is given by the formula 
  $$D(n,m)=\frac{n+1-2\cdot m}{n+1-m}\cdot{n \choose m}$$ 
  and satisfies the recurrence relation 
  $$D(n,m)=D(n-1,m)\:+\:\sum_{i=0}^{m-1} D(2\cdot i,i)\cdot 
  D(n-2\cdot(i+1),m - (i+1)).$$ 
  Obviously, if $n = 2 \cdot m$, then we obtain the recurrence 
  relation for the Catalan numbers (starting from $0$) 
  $$C_{m+1}=\sum_{i=0}^{m-1} C_{i+1}\cdot C_{m-i}.$$ 
  Using the above recurrence relation we can see 
  that $$\sum_{i=0}^\infty\,C_{i+1}
  \cdot x^i\,= \,1\,+\,\left(\sum_{i=0}^\infty\, C_{i+1}\cdot x^i\right)^2$$ 
  where ($| x |< \frac{1}{4}$) and hence 
  $$\sum_{i=0}^\infty\, C_{i+1}\cdot x^i\,= \,
  \frac{1-\sqrt[]{1-4\cdot x}}{2\cdot x}.$$}}

