On the construction of asymptotic expansion of high precision

We consider the new asymptotic expansions in the local forms of the central limit theorem. If the initial independent identically distributed random variables have unit variance and finite sixth moment, some of these expansions provide accuracy of approximations of distribution of normalized sums close to

$$\frac{\beta_6}{120n^2},$$

where n number of summands. We also give numerical illustrations of these results.