Proof of a conjecture of Mircea Merca

We prove that, for any prime p and positive integer r with pr>2, the number of multinomial coefficients such that(kk1,k2,…,kn)=pr,andk1+2k2+⋯+nkn=n, is given byδpr,k(⌊n−1pr−1⌋−δ0,nmodpr), where δi,j is the Kronecker delta and ⌊x⌋ stands for the largest integer not exceeding x. This confirms a recent conjecture of Mircea Merca.