My edits to OEIS sequences A001299 and A001300 were recently approved. I added the article “Partitions in Monetary Units” by John Riordan in the references section and I also added formulas from the article. The article can be found in Scripta Mathematica, Vol. 6 (1939), pp. 182-183. The article is also relevant to sequence A187243, but I have not submitted any edits on the sequence’s page yet.
This gives me an opportunity to encourage other people to add formulas and references from older books or math journals to OEIS. Sometimes, you even find sequences that are not in the database like A383873. I found the sequence A383873 in “A Mathematical and Philosophical Dictionary, Vol 2.” by Charles Hutton, which was published in late 18th century. Charles Hutton’s book was also relevant for sequences A000578, A014481 and A224446. I am not even done yet with Hutton’s book when it comes to OEIS sequences. The journal Scripta Mathematica also has a few articles and formulas relevant to various sequences.
New Formulas and Whittaker’s Root Series Formula
I believe that Whittaker’s Root Series Formula can be used to obtain new formulas for A001299, A001300 and A187243. In this case, it’s only the denominator part of the formula that is relevant.
In a previous post entitled “A Note on Whittaker’s Root Series Formula” I discussed how the denominator part of Whittaker’s Root Series Formula is related to the series expansion of 1/p(x), where p(x) is a polynomial p(x)=anxn+an-1xn-1+…+a1x+a0. The generating functions for the 3 sequences have the 1/p(x) form, so the ideas from my previous post are relevant. The only problem is that I only discussed polynomials that have a0 =1. The generating functions for the 3 sequences don’t meet this condition, but I don’t think this is a major problem. When a0 is not 1, we probably just have to divide by a constant to match the coefficients of the expansion.
So with Whittaker’s Root Series Formula, I should be able to obtain the terms of the sequences with the help of determinants of Toeplitz matrices. In the end, these Toeplitz matrices should follow a recursive formula where the next term is based on previous terms. A187243 deals with a third degree p(x), so it should be the easiest.
Right now I may not have the time to work on these formulas. Anybody that is interested could try themselves to follow my ideas and if you find new formulas you should publish them on OEIS. If you need some guidance you can comment here or you can send me an email. In the end I want to encourage other people to do interesting OEIS edits ,to learn about Whittaker’s Root Series Formula or to learn about other topics covered by my blog.