The word Akh (אך - only) in the phrase והיית אך שמח "you shall be only happy" (Debarim 16:15), has a numerical value of 21. This comes to tell us that we must be happy till the 21st day of Tashri, which is the last day of Sukkoth, which is Hosha'na Rabba.
The Mishnah Berurah states that it is the custom to be awake on the night of Hosha'na Rabba doing the published Seder. I have always wondered why it is that the majority of Ashkenazim do not follow the Mishnah Berurah in this matter. The custom of staying up all night on Hosha'na Rabba is widespread, however, among Sephardim.
As mentioned previously, it is appropriate for women to also stay up on this night, wherever possible, since the Tiqqun applies to them also. This is in contrast to the reason for staying up on the night of Shabu'oth, which is not relevant to women.
(See Rabben Bahya on the torah, Debarim 16:15. Mishnah Berurah, 664:1. Ben Ish Hai, 1st Year, Wezoth HaBerakha, Oth 1-3)
Gematria, Hoshana Rabbah, women stay up all night, Tishrei, Shavuot