I'm interested in polynomials of the following form:
$$p_k(x)=\prod_{j=1}^k(x+j)^{\lfloor{\frac{k}{j}\rfloor}{}}$$
Outside any of my own analysis, I wondered if there is any name for these polynomials, if they have been studied before, if there are any papers written about properties of these polynomials.
EDIT: Note that I wrote the original exponent incorrectly. It should have been $k/j$ in the floor function
$$p_1(x)=(x+1)$$ $$p_2(x)=(x+1)^2(x+2)$$ $$p_3(x)=(x+1)^3(x+2)(x+3)$$ $$p_4(x)=(x+1)^4(x+2)^2(x+3)(x+4)$$ $$p_5(x)=(x+1)^5(x+2)^2(x+3)(x+4)(x+5)$$ $$p_6(x)=(x+1)^6(x+2)^3(x+3)^2(x+4)(x+5)(x+6)$$