As a part of one computer algorithm, I want to find sum for $$n+ \frac{n(n+1)}{2!} + \frac{n(n+1)(n+2)}{3!}+....+ \frac{n(n+1)(n+2)...(n+r-1)}{r!} $$. I looked at $$\frac1{(1-x)^n}$$. But it is infinite series. Is there some formula? Please help.
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Your series is $\sum_{k=1}^r\binom{n-1+k}k$. – Brian M. Scott Apr 10 '15 at 18:36
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$$\sum_{r=0}^{n}\binom{n+r-1}r=\sum_{r=0}^{n}\binom{n+r-1}{n-1}=\binom{2n}n$$ – Hypergeometricx Apr 11 '15 at 06:28
