The coefficients of the multinomial formula are integers where the numerator is the factorial of a sum of integers and the denominator is the product of the factorials of those same integers.
Suppose instead of taking a product of those factorials in the denominator we took the sum of those factorials. When would that expression be an integer?
Based on some calculations, I suspect, but I cannot prove in general, that if the integers are consecutive forming the sum, then there exists a non-negative integer $m$ depending on the number of integers $k$ in the sum such that if $n \ge m$ then the expression is always an integer.
In other words, there exists $m$ depending on $k$ such that for all $n \ge m$ the following is always an integer:
For $k = 1, m = 0$. For $k = 2, m = 1$. Based on some calculations, I conjecture, but I have no proof, for $k = 3, m = 1$ and for $k = 4, m = 4$.
Is this type of problem familiar to anyone? Perhaps someone has a solution to it or suggestions where I could go for more information.