In this paper we compute the coefficients of the reliability polynomial of a consecutive-$k$-out-of-$n$:$F$ system, in Bernstein basis, using the generalized Pascal coefficients. Based on well-known combinatorial properties of the generalized Pascal triangle we determine simple closed formulae for the reliability polynomial of a consecutive system for particular ranges of $k$. Moreover, for the remaining ranges of $k$ (where we were not able to determine simple closed formulae), we establish easy to calculate sharp bounds for the reliability polynomial of a consecutive system.

中文翻译：

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连续 $k$-out-of-$n$:$F$ 系统的系数的严格界限

在本文中，我们使用广义帕斯卡系数以 Bernstein 为基础计算连续-$k$-out-of-$n$:$F$ 系统的可靠性多项式的系数。基于广义帕斯卡三角形的众所周知的组合特性，我们为特定范围的 $k$ 的连续系统的可靠性多项式确定了简单的闭合公式。此外，对于 $k$ 的剩余范围（我们无法确定简单的封闭公式），我们为连续系统的可靠性多项式建立了易于计算的锐界。