Complete bell polynomial coefficients

Question

I would like to know if it is possible to calculate the coefficient of a given Complete Bell Polynomial 's monomial by its indexes and powers:

$B_{n}(x_1,x_2,...,x_n)= c_n(1,n) x_1^n + c_n((1,n-2),(2,1)) x_1^{n-2}x_2 +...+c_n((1,j_1),(2,j_2),...(n,j_n))x_1^{j_1} x_2^{j_2}...x_n^{j_n} $

I want a formula for $c_n((1,j_1),(2,j_2),...(n,j_n))$.

I want a way to calculate that the coefficient of the monomial $x_1x_2^3$ of $B_7(x_1,...x_7)$ is 105, with a formula similar to $c_7((1,1),(2,3))=\frac{7!}{1!^{1}2!^3 3!}$

Answer 1

It looks like the example of formula that I give is the actual Formula.

$c_n((1,j_1),(2,j_2),\dots ,(n,j_n))=\frac{n!}{j_1!j_2!\dots j_n!1!^{j_1}2!^{j_2}\dots n!^{j_n}}$