What are the coefficients in the expansion of $(x+y)(x+2y) \cdots (x+ny)$? Are the numbers appearing as coefficients in the following sequence of polynomials known? Is there a known recurrence relation to compute them?
\begin{align*}
  (x+y) &= x+y \\
  (x+y)(x+2y) &= x^2+3yx+2y^2 \\
  (x+y)(x+2y)(x+3y) &= x^3+6yx^2+11y^2x+6y^3 \\
  (x+y)(x+2y)(x+3y)(x+4y) &= x^4+10yx^3+35y^2x^2+50y^3x+24y^4 \\
 &\text{etc.}
\end{align*}

The On-Line Encyclopedia of Integer Sequences® is a
searchable database of many integer sequences, and (as @Kevin also said in a comment) can be a valuable
tool to identify a particular sequence. Searching for
$$
    1, 1, 1, 3, 2, 1, 6, 11, 6, 1, 10, 35, 50, 24
$$
returns as the top result

A094638 Triangle read by rows: T(n,k) = |s(n,n+1-k)|, where s(n,k) are the signed Stirling numbers of the first kind (1 <= k <= n; in other words, the unsigned Stirling numbers of the first kind in reverse order).

So your numbers are Unsigned Stirling numbers of the first kind ${ n \brack k }$, which can be defined as the coefficients in the expansion 
$$ \tag{*}
x(x+1)(x+2) \cdots (x+n-1) = \sum_{k=0}^n { n \brack k } x^k
$$
They satisfy various recurrence relations, such as
$$
{n+1 \brack k} = n { n \brack k } + { n \brack k-1 }
$$
for $k > 0$, with the initial conditions
$$
 { 0 \brack 0 } = 1 \, , \quad { n \brack 0 } = { 0 \brack n } = 0
$$
for $n > 0$. For more resources about the Stirling numbers and their
relations, see @Michael's comment below.

The precise connection with your polynomial expansion is
$$
 (x+y)(x+2y) \cdots (x+ny) = \sum_{k=0}^n { n+1 \brack k+1 } x^k y^{n-k}
$$
This follows from $(*)$ with $n+1$ instead of $n$ and $\frac xy$ instead of $x$.
