Properties of sequence

In what follows, I will denote the ratio you mentioned $\binom{n}{k}_s$, in analogy with the binomial coefficients. I do not know of a nice characterization of all sequences for which this property holds, but here are a few of infinite families of examples.

• Let $a$ and $b$ be integers, and let $s_n$ satisfy the recurrence relation $$ s_n=as_{n-1}+bs_{n-2},\\ s_1=s_2=1\qquad\;\;\;\;\; $$ Then the ratio is always an integer. When $a=b=1$, then $s_n$ is the sequence of Fibonacci numbers, and the $\binom{n}{k}_s$ are called the Fibinomial coefficients.

• Let $f$ be any polynomial with integer coefficients, and let $x_0$ be any fixed integer. Then another sequence which works is $$s_n=(\underbrace{f\circ f\circ\dots\circ f}_{\text{$n$ times}})(x_0)-x_0,$$ where $f$ is composed with itself $n$ times (Proof). The most famous example in this family is when $f(x)=qx$ and $x_0=1$, for some $q\neq 1$, in which case $s_n=q^n-1$, and the numbers $\binom{n}{k}_s$ are called the $q$-binomial coefficients. The same coefficients appear when $f(x)=qx+b$. For $f(x)=x+b$, the usual binomial coefficients appear. When $\deg f\ge 2$, $\binom{n}{k}_s$ grows quite quickly.

• If $d$ is any fixed positive integer, then $s_n=n(n+1)\cdots (n+d-1)$ works. When $d=2$, these are the Narananya numbers. For general $d$, $\binom{n}{k}_s$ is the number of plane partitions which fit in a $k\times (n-k)\times d$ box. Equivalently, this is the number of tilings of an equiangular hexagon with side lengths $k,n-k,d,k,n-k,d$ by rhombi with side length $1$.

• One final mention: Knuth and Wilf prove in The power of a prime that divides a generalized binomial coefficient that as long as the sequence $s$ satisfies the following property, then $\binom{n}{k}_s$ will always be an integer: $$ \gcd(s_n,s_m)=s_{\gcd(n,m)} $$