I came across a very well written article named Partnership, Partition, and Proof: The Path to the Hardy Ramanujan Partition Formula. One of the author Adrian Rice’s Major point (pun intended) was that MacMahon’s manual computation of p(200) which concurred with Hardy and Ramanujan’s estimate was a big driving force behind the completion of their work. Hardy and Ramanujan’s acknowledgement is quoted below.
To Major MacMahon in particular we owe many thanks for the amount of trouble he has taken over very tedious calculations. It is certain that, without the encouragement given by the results of these calculations, we should never have attempted to prove theoretical results at all comparable in precision with those which we have enunciated.
Nowadays instead of the services of Major MacMahons, we use Maple MatheMAtica. Interestingly, if you have watched “The Man Who Knew Infinity”, the relation between Hardy and Ramanujan, and MacMahon was portrayed as adversarial. Someone actually put the whole movie up on youtube. This part can be found here from 21:30 onwards to about 27:00. From another paper by Rice, Partnership and Partition: A Case Study of Mathematical Exchange, it seems that although MacMahon was made a member of St John’s college in 1904 but he only moved to Cambridge in 1922. In retrospect, perhaps this was the reason that Ramanujan was only introduced to MacMahon at such a late stage in the movie. Since we are discussing Ramanujan versus MacMahon (real or reel) it is fitting to read Andrews’ review of the movie in the notices of AMS, where he discusses the meeting in the movie between the two protagonists and gives a nice quote of Rota.