You could also do this using perl's Net::FTP module:

#!/usr/bin/perl
use strict;
use warnings;
use Net::FTP;

#Change your.ftp-host.com to your ftp domain
my $ftp = Net::FTP->new("your.ftp-host.com", Debug => 0)
        or die "Can not connect: $@";
#Change UserName and PassWord to your login information
$ftp->login("UserName",'PassWord')
        or die "Can not log in: ", $ftp->message;
#If you want to list your root directory you can comment this out
#Or change public_html to where you want to get a file list
$ftp->cwd("public_html");

my @list = $ftp->ls($ftp->pwd);
my $listing = "list.txt";

open(LISTING, ">$listing");

foreach my $line (@list) {
        print LISTING $line . "\n";
}

close(LISTING)