Sum of the ratios of binomial coefficients

Asked Jan 01 '21 at 13:00

Active Jan 01 '21 at 13:11

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How can I find the sum of the following ratio of binomial coefficient $$ \sum_{j=k}^n \frac{ {j \choose k} }{ {n \choose j}} $$

I found a question similar to mine but differs in the summation index. Is there a general strategy for finding the sum of binomial coefficients?

edited Jan 01 '21 at 13:11

Bernard

asked Jan 01 '21 at 13:00

Shew

1

For $k=0$, see Calculate sums of inverses of binomial coefficients. – Vepir Jan 01 '21 at 14:02
This seems like a job for Zeilberger algorithm. – Phicar Jan 01 '21 at 18:10
It only finds 2nd-order recurrence (w.r.t. $n$ or $k$, separately). – metamorphy Jan 07 '21 at 12:26

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