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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- Jun 5th 2012, 01:33 AMobergurusum involving binomials
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?

- Jun 5th 2012, 03:32 AMtom@ballooncalculusRe: sum involving binomials
[tex] for $

$\displaystyle \displaystyle \begin{align*} \\ \sum_{s=t}^{t+d}\binom{s}{t}\frac{1}{s+1}\ &=\ \sum_{s=0}^{d}\binom{t + s}{t}\frac{1}{t + s + 1} \\ \\ &= \ \sum_{s=0}^{d}\frac{(t+s)!}{t!\ s!\ (t + s + 1)} \end{align*}$

sum from s=0 to s=d of (t + s)! / [t! s! (t + s + 1)] - Wolfram|Alpha

Closed-form expression - Wikipedia, the free encyclopedia - Jun 5th 2012, 05:33 AMoberguruRe: sum involving binomials
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 :(