binomial series sum

[jacks]

$$ \sum_{0 \leq j <i \leq n}  j{n \choose i}$$

[el_manco]

Hola

$$ \sum_{0 \leq j <i \leq n}  j{n \choose i}$$

This is:

[texx]\displaystyle\sum_{i=0}^n{}{n \choose i}\displaystyle\sum_{j=0}^{i-1}{}j=\displaystyle\sum_{i=0}^n{}{n \choose i}\dfrac{i(i-1)}{2}[/texx]

Now note that:

[texx]f(x)=(1+x)^n=\displaystyle\sum_{i=0}^n{}\displaystyle\binom{n}{i}x^i[/texx]

Compute [texx]f''(1)[/texx].

Best regards.

[jacks]

Thanks Admin for nice solution