Quaestiones Mathematicae ( IF 1.049 ) Pub Date : 2020-10-16 , DOI: 10.2989/16073606.2020.1825019 Mircea Merca
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.