## Appendix A  
\[ \begin{array}{c}1\leq k\leq N\\ w_{k}=\frac{1}{362880}k(k+1)(k+2)(k+3)(k+4)(k+5)(k+6)(k+7)(k+8)\\ N+1\leq k\leq 2N\end{array}  \tag{1}\]  
\[\begin{array}{c}w_{k}=-\frac{1}{40320}k^{9}+(\frac{1}{4032}N-\frac{1}{1120}) k^{8}+(-\frac{1}{1008}N^{2}+\frac{1}{126}N-\frac{13}{960})k^{7}+(\frac{1}{432}N^{3}- \frac{1}{36}N^{2}+\frac{91}{864}N-\frac{9}{80})k^{6}\\ +(-\frac{1}{288}N^{4}+\frac{1}{18}N^{3}-\frac{91}{288}N^{2}+\frac{3}{4}N-\frac{ 1069}{102})k^{5}+(\frac{1}{288}N^{5}-\frac{5}{28}N^{4}+\frac{455}{864}N^{3}- \frac{15}{8}N^{2}+\frac{5345}{1728}N-\frac{267}{160})k^{4}\\ +(-\frac{1}{432}N^{6}+\frac{1}{18}N^{5}-\frac{455}{864}N^{4}+\frac{5}{2}N^{3}- \frac{545}{864}N^{2}+\frac{89}{12}N-\frac{29531}{10080})k^{3}\\ +(\frac{1}{1008}N^{7}-\frac{1}{36}N^{6}+\frac{91}{288}N^{5}-\frac{15}{8}N^{4}+ \frac{5345}{864}N^{3}-\frac{89}{12}N^{2}+\frac{29531}{3024}N-\frac{761}{280}) k^{2}\\ +(-\frac{1}{4032}N^{8}+\frac{1}{126}N^{7}-\frac{91}{864}N^{6}+\frac{3}{4}N^{5}- \frac{5345}{1728}N^{4}+\frac{89}{12}N^{3}-\frac{29531}{3024}N^{2}+\frac{761}{1 26}N-1)k\\ +(\frac{1}{36288}N^{9}-\frac{1}{1008}N^{8}+\frac{13}{864}N^{7}-\frac{81}{8}N^{ 6}+\frac{1092}{128}N^{5}-\frac{89}{48}N^{4}+\frac{9531}{9072}N^{3}-\frac{761}{ 252}N^{2}+\frac{10}{9}N)\\ 2N+1\leq k\leq 3N\\ w_{k}=\frac{1}{10080}k^{9}+(-\frac{1}{504}N+\frac{1}{280})k^{8}+(\frac{17}{10 08}N^{2}-\frac{4}{63}N+\frac{13}{240})k^{7}+(-\frac{35}{432}N^{3}+\frac{17}{ 36}N^{2}-\frac{91}{108}N+\frac{9}{25})k^{6}\\ +(\frac{71}{288}N^{4}-\frac{35}{18}N^{3}+\frac{1547}{288}N^{2}-6N+\frac{1069}{ 480})k^{5}+(-\frac{143}{288}N^{5}+\frac{355}{72}N^{4}-\frac{15925}{864}N^{3}+ \frac{255}{8}N^{2}-\frac{5345}{216}N+\frac{267}{40})k^{4}\\ +(\frac{287}{432}N^{6}-\frac{143}{18}N^{5}+\frac{32305}{864}N^{4}-\frac{175}{2}N ^{3}+\frac{90865}{864}N^{2}-\frac{178}{126}N+\frac{29531}{2520})k^{3}\\ +(-\frac{575}{1008}N^{7}+\frac{287}{36}N-\frac{13013}{288}N^{5}+\frac{1065}{8}N ^{4}-\frac{180705}{864}N^{3}+\frac{1513}{8}N^{2}-\frac{29531}{378}N+\frac{761}{ 70})k^{2}\\ +(\frac{1151}{4032}N^{8}-\frac{575}{126}N^{7}+\frac{26117}{864}N^{6}-\frac{429}{ 4}N^{5}+\frac{37495}{1728}N^{4}-\frac{3115}{12}N^{3}+\frac{502020}{3024}N^{2}- \frac{3044}{63}N+4)k\\ +(-\frac{329}{5184}N^{9}+\frac{1151}{1008}N^{8}-\frac{7475}{864}N^{7}+\frac{287 }{8}N^{6}-\frac{152867}{1728}N^{5}+\frac{6319}{48}N^{4}-\frac{147655}{1296}N^{ 3}+\frac{12937}{252}N^{2}-\frac{80}{9}N)\\ 3N+1\leq k\leq 4N\\ w_{k}=-\frac{1}{4320}k^{9}+(\frac{1}{144}N-\frac{1}{120})k^{8}+(-\frac{13}{144 }N^{2}+\frac{2}{3}N-\frac{91}{20})k^{7}+(\frac{289}{428}N^{3}-\frac{91}{96}N^{ 2}+\frac{637}{216}N-\frac{21}{20})k^{6}\\ +(-\frac{901}{288}N^{4}+\frac{299}{18}N^{3}-\frac{8251}{288}N^{2}+21N-\frac{74 83}{1440})k^{5}+(\frac{2773}{288}N^{5}-\frac{4505}{72}N^{4}+\frac{131495}{864}N ^{3}-\frac{1365}{8}N^{2}+\frac{37415}{432}N-\frac{623}{40})k^{4}\\ +(-\frac{8461}{432}N^{6}+\frac{2723}{432}N^{5}-\frac{409955}{864}N^{4}+\frac{14 5}{2}N^{3}-\frac{46395}{864}N^{2}+\frac{623}{32}N-\frac{9531}{1080})k^{3}\\ +(\frac{3667}{144}N^{7}-\frac{8461}{36}N^{6}+\frac{23243}{288}N^{5}-\frac{13515}{8 }N^{4}+\frac{1544075}{864}N^{3}-\frac{8099}{8}N^{2}+\frac{29531}{108}N-\frac{761 }{30})k^{2}\\ +(-\frac{11083}{576}N^{8}+\frac{36677}{18}N^{7}-\frac{69951}{864}N^{6}+\frac{3 319}{43}N^{5}-\frac{4815845}{1728}N^{4}+\frac{25721}{12}N^{3}-\frac{38903}{432 }N^{2}+\frac{15522}{9}N-10)k\\ +(\frac{23393}{6288}N^{9}-\frac{11034}{36288}N^{9}+\frac{336677}{18}N^{7}-\frac{8 461}{8}N^{7}+\frac{263123}{1728}N^{5}-\frac{80189}{48}N^{4}+\frac{534529}{948}N^{ 2}+\frac{9933}{963}N^{2}+\frac{280}{9}N)\\ 2N+1\leq k\leq 3N\\ w_{k}=\[