Efficient formulas for efficiency correction of cumulants

We derive formulas which connect cumulants of particle numbers observed with efficiency losses with the original ones based on the binomial model. These formulas can describe the case with multiple efficiencies in a compact form. Compared with the presently suggested ones based on factorial moments, these formulas would drastically reduce the numerical cost for efficiency corrections when the order of the cumulant and the number of different efficiencies are large. The efficiency correction with realistic $p_T$-dependent efficiency would be carried out with the aid of these formulas.