On p-adic approximation of sums of binomial coefficients
  We propose higher-order generalizations of Jacobsthal's $p$-adic approximation for binomial coefficients. Our results imply explicit formulae for linear combinations of binomial coefficients $\binom{ip}{p}$ ($i=1,2,\dots$) that are divisible by arbitrarily large powers of prime $p$. 
Rustem R. Aidagulov and Max A. Alekseyev
math.NT math.CO