While the link provided by tagteam references a due date, the date has not been updated to reflect the 2016 filing season.  However, because due date is April 17th and this is Emancipation Day, the due date for the 2016 form is April 18, 2017.

The link should also be sufficient to show the beneficiaries that the due date is in April and not January.

This assumes the tax year is a calendar year.

To add a bit of extra info, for calendar year estates and trusts, Form 1041 and Schedule(s) K-1 are due on or before April 18, 2017.  For fiscal year estates, the due date is the 15th day of the 4th month following the close of their tax year. For example, an estate that has a tax year that ends on June 30, 2017, must file Form 1041 by October 16, 2017. If the due date falls on a Saturday, Sunday, or legal holiday, the due date is the next business day.
And the beneficiaries WILL need the K-1 to complete their taxes.

To add even more info, the aforementioned due dates presume an extension has not been filed by the fiduciary, which would also extend the due date for providing K-1(s) to the beneficiary(ies).