Binomial series sum

[man111]

(1) [tex]\displaystyle \sum_{0\leq i < j \leq n}j\binom{n}{i}[/tex]

(2) [tex]\displaystyle \sum_{0 \leq j \leq i \leq n} {n \choose i} {i \choose j}[/tex]

[man111]

(1) [tex]\displaystyle \sum \sum_{0\leq i < j \leq n}j\binom{n}{i}[/tex]

(2) [tex]\displaystyle \sum\sum_{0 \leq j \leq i \leq n} {n \choose i} {i \choose j}[/tex]

[man111]

please have a look on above problems, thanks

[man111]

For (2) [tex]\displaystyle S = ^nC_{1}\bigg( ^1 C_{0} \bigg)+^nC_{2}\bigg(^2C_{0}+^2C_{1}\bigg)+\cdots +^nC_{n}\bigg(^nC_{0}+^nC_{1}+^nC_{2}+\cdots \bigg)[/tex]

[tex]\displaystyle S=^nC_{1}(2-1)+^nC_{2}(2^2-1)+\cdots ^nC_{n}(2^n-1)=3^n-2^n[/tex]

[man111]

For (1) [tex]\displaystyle S = 1(^nC_{0})+2(^nC_{0}+^nC_{1})+3(^nC_{0}+^nC_{1}+^nC_{2})+\cdots +n(^nC_{0}+^nC_{1}+\cdots +^nC_{n-1})[/tex]

[tex]\displaystyle S = (1+2+\cdots +n)^nC_{0}+(2+3+\cdots+n)^nC_{1}+(3+4+\cdots +n)^nC_{2}+.. +n(^nC_{n-1})[/tex]

[tex]\displaystyle S =\sum^{n}_{i=0}\binom{n}{i}\bigg[\frac{n(n+1)}{2}-\frac{i(i+1)}{2}\bigg][/tex]

[tex]\displaystyle S = \sum^{n}_{i=0}\binom{n}{i}\frac{(n-i)(n+i+1)}{2}=\frac{n}{2}\sum^{n}_{i=0}\binom{n-1}{i}(n+i+1)[/tex]

[tex]\displaystyle S=\frac{n^2+1}{2}\sum^{n}_{i=0}\binom{n-1}{i}+\frac{n^2}{2}i\binom{n-1}{i}[/tex]

Using [tex]\displaystyle (1+x)^{n-1}=\sum^{n-1}_{i=0}\binom{n-1}{i}x^i,[/tex] put [tex]x=1,[/tex] Then [tex]\displaystyle 2^{n-1}=\sum^{n}_{i=0}\binom{n}{i}[/tex]

[tex]\displaystyle (n-1)(1+x)^{n-2} = \sum^{n-1}_{k=0}\binom{n-1}{i}ix^{i-1},[/tex] put [tex]x=1,[/tex] Then [tex]\displaystyle (n-1)2^{n-2} = \sum^{n-1}_{i=0}i\binom{n-1}{i}[/tex]

So [tex]\displaystyle S = \frac{n^2+1}{2}\cdot 2^{n-1}+\frac{n^2}{2}\cdot (n-1)2^{n-2}[/tex]