I have a particular polynomial
$$ 1-10x+35x^2-50x^3 $$
Which can be written nicely as
$$1-(1+2+3+4)x+(1\cdot2+1\cdot3+1\cdot4+2\cdot3+2\cdot4+3\cdot4)x^2$$ $$+(1\cdot2\cdot3+1\cdot2\cdot4+1\cdot3\cdot4+2\cdot3\cdot4)x^3$$
I can write the coefficients as a pattern using summation notation as
$$\sum_{1\le j_1<...<j_4\le4}\frac{4!}{j_1\cdot...\cdot j_4}-\sum_{1\le j_1<j_2<j_3\le4}\frac{4!}{j_1\cdot j_2\cdot j_3}x+\sum_{1\le j_1<j_2\le4}\frac{4!}{j_1\cdot j_2}x^2-\sum_{1\le j_1\le4}\frac{4!}{j_1}x^3$$
I would like to condense this down in to a single double summation. I think I can handle it by doing
$$4!\sum_{k=1}^{4}\sum_{1\le j_1\lt ...j_{5-k}\le 4}\frac{(-x)^{k-1}}{j_1\cdot...\cdot j_k}$$
a) Is this notation okay? What i mean is, it the notation correct mathematically?
b) Is there an easier way to write this?
EDIT: I made a slight error in the denominator. I think this fixes is.
$$4!\sum_{k=1}^{4}\sum_{1\le j_1\lt ...j_{5-k}\le 4}\frac{(-x)^{k-1}}{j_1\cdot...\cdot j_{5-k}}$$
EDIT 2: I came across the generalized harmonic numbers defined as $$H_{n,j}=\sum_{1\le k_1<...,<k_j\le n}\frac{1}{k_1\cdot...\cdot k_j}$$
Therefore I can write the above double sum using this definition as
$$4!\sum_{j=0}^3H_{4,4-j}x^j$$