From this question and the solution here a nice closed form has been provided for the sum of a lacunary series of binomial coefficients as follows:
$$\sum_{j=0}^{n} \binom{n}{dj} = \frac{1}{d} \sum_{j=0}^{d-1} (1 + \omega^j)^n$$ where $\omega$=primitive $d$-th root of unity.
The question here is whether there is a closed form (or a form similar to the above) if the lacunary series is of alternating sign, i.e. $$\sum_{j=0}^n \binom n{dj}(-1)^j$$
* Additional Note *
Perhaps the upper limit of the original summation should be $\big\lfloor\frac nd\big\rfloor$.