Is there a closed formula for the following sum involving binomials: $\sum_{s=t}^{t+d}\binom{s}{t}\frac{1}{s+1}$ where t,d>=0?
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$\displaystyle \displaystyle \begin{align*} \\ \sum_{s=t}^{t+d}\binom{s}{t}\frac{1}{s+1}\ &=\ \sum_{s=0}^{d}\binom{t + s}{t}\frac{1}{t + s + 1} \\ \\ &= \ \sum_{s=0}^{d}\frac{(t+s)!}{t!\ s!\ (t + s + 1)} \end{align*}$
sum from s=0 to s=d of (t + s)! / [t! s! (t + s + 1)] - Wolfram|Alpha
Closed-form expression - Wikipedia, the free encyclopedia
gr8 many thanks, tom. but i'm looking for a formula that does not involve sums or (unresolved) hypergeometric series moreover, the wolfram solution contains the value of the Gamma function for negative integers