[
ewcommand{e}{mathrm{e}}
egin{aligned}
leftlangleegin{matrix}n \ iend{matrix}
ight
angle&=sum_{j=i}^{n-1}(-1)^{j-i} inom ji imes n ^ {n - j}\
&=sum_{j=i}^{n-1}(-1)^{j-i}inom ji imes n! [x^n] sum_{k=0}^{n-j}e^{kx}(-1)^k inom{n-j}k \
&=sum_{j=i}^{n-1}(-1)^{j-i}inom ji sum_{k=0}^{n-j} k^n (-1)^{n-j-k} inom{n-j}k\
&=sum_{k=1}^{n-i} k^n (-1)^{n-i-k} sum_{j=0}^{n-k} inom{n-j}k inom ji \
&=sum_{k=1}^{n-i} k^n (-1)^{n-i-k}inom{n+1}{k+i+1} \
&=sum_{k=1}^{i+1} k^n (-1)^{n-(n-1-i)} inom{n+1}{k+(n-1-i)+1} \
&=sum_{k=1}^{i+1} k^n (-1)^{i+1-k} inom{n+1}{i+1-k} \
&=sum_{k=0}^i (k+1)^n (-1)^{i-k} inom{n+1}{i-k} \
&=sum_{k=0}^i (i+1-k)^n (-1)^k inom{n+1}{k}
end{aligned}
]