The New York City Dept. of Records lets you request a search for death
records based on date of death and name alone. You wouldn't need the
certificate number. I think it costs about $20. I've ordered records >from
them before. It takes about 6 weeks to get a response but the response
does come.

http://www.nyc.gov/dorforms/death.jsp. Also, Steve Morse has a useful
page for searching NYC death records. Morse's site searches the
italiangen.org website, so if the NYC record you're looking for
exists and is mis-indexed on italiangen, you might not find it on Morse's
site either.

http://stevemorse.org/vital/nydeath.html

Dave Strausfeld