Find the exponential generating function for the number of alphanumeric strings of length n formed from the 26 uppercase letters of the English alphabet and 10 decimal digits if:
-each vowel must appear at least one time;
-the letter T must appear at least three times;
-the letter Z may appear at most three times;
-each even digit must appear an even number of times; and
-each odd digit must appear an odd number of times.