Mean divisibility of multinomial coefficients

Let m1,…,ms be positive integers. Consider the sequence defined by multinomial coefficients:an=((m1+m2+⋯+ms)nm1n,m2n,…,msn). Fix a positive integer k⩾2. We show that there exists a positive integer C(k) such that∏n=1takn∏n=1tan∈1C(k)Z for all positive integer t, if and only if GCD(m1,…,ms)=1.