The Greatest Common Divisor of a Binomial Row

Write h(n) = (n1, , nn-1). The answer is governed entirely by carries in base-p addition: by Kummer's theorem the exponent of p in ni counts the carries when i and $n-i$ are added in base p.From that criterion, $h(n) = p$ when n is a power of the prime p, and $h(n) = 1$ otherwise. The intermediate observation worth isolating is that divisibility at a single index proves nothing: whenever p n the index $i = p-1$ forces a carry out of the units place, so p np-1 always. Only a prime dividing the whole row contributes to the gcd.