# 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
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 $$x$$.