In continuation to my question posted yesterday ,I obtained one table on the basis of some initial values, and tried to observe some pattern out of it.
\begin{array}{|c|c|c|c|} \hline 0& & & & & & & & & & & & & & & \\ \hline 0&0 & & & & & & & & & & & & & &\\ \hline 0 &0 & 0& & & & & & & & & & & & &\\ \hline 0& 0& 1& 1& & & & & & & & & & & &\\ \hline 0& 0& 3&5 &1 & & & & & & & & & & &\\ \hline 0& 0& 7&14 &6 &1 & & & & & & & & & &\\ \hline 0&0 &13 &32 &21 &7 &1 & & & & & & & & &\\ \hline 0& 0& 22&63 &56 &28 &8 &1 & & & & & & & &\\ \hline 0& 0& 34&112 &126 &84 &36 &9 &1 & & & & & & &\\ \hline 0& 0& 50&185 &251 &210 &120 &45 &10 &1 & & & & & &\\ \hline 0& 0& 70&289 &459 &462 &330 &165 &55 &11 &1 & & & & &\\ \hline 0& 0& 95& 431& 785&924&792 &495 &220 &66 &12 &1 & & & &\\ \hline 0& 0&125 &620 &1273 &1716 &1716 &1287 &715 &286 &78 &13 &1 & & &\\ \hline 0& 0& 161& 865& 1976& 3003&3432 &3003 &2002 &1001 &364 &91 &14 &1 & &\\ \hline 0& 0& 203&1176 &2959 &5004 &6435 &6435 &5005 &3003 &1365 &455 &105 &15 &1 &\\ \hline 0&0 &252 &1564 &4297 &8005 &11440 &12870 &11440 &8008 &4368 &1820 &560 &120 &16 &1\\ \hline \end{array}
Based on the above mentioned table , i observed that as we move towards right half of table f(i,j)=f(i-1,j-1)+f(i-1,j) . Can someone help me out using this to derive any recurrence if it exists?