Abstract

In this paper, we investigate two methods to express the natural powers of 2 as sums over integer partitions. First we consider a formula by N. J. Fine that allows us to express a binomial coefficient in terms of multinomial coefficients as a sum over partitions. The second method invokes the central binomial coefficients and the logarithmic differentiation of their generating function. Some experimental results suggest the existence of other methods of decomposing the power of 2 as sums over partitions.

中文翻译：

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两个的幂作为分区的总和

摘要

在本文中，我们研究了两种将 2 的自然幂表示为整数分区上的和的方法。首先，我们考虑 NJ Fine 的一个公式，它允许我们根据多项式系数将二项式系数表示为分区的总和。第二种方法调用中心二项式系数及其生成函数的对数微分。一些实验结果表明存在将 2 的幂分解为分区总和的其他方法。