1

How can I simplify $ \frac { \sum_{n=0}^{ \infty } \frac {(-1)^{n-1}n^{n-1}x^{n}} {n!}} {x\sum_{n=0}^{ \infty }\frac{x^n}{n!} }$ so that I can evaluate it?

  • If you can't do it to a ratio of two polynomials, what makes you think you can do it to a ratio of two infinite series? – J. M. ain't a mathematician Mar 28 '19 at 13:51
  • 2
    The denominator is simply $x e^x$. As for the numerator, I am pretty sure that the sum begins with $n = 1$, since the term does not make sense for $n=0$. Anyway, in such case, the numerator is simply $W_0(x)$, the Lambert W-function. Summarizing, the expression simplifies to $$ \frac{\sum_{n=1}^{\infty} \frac{(-n)^{n-1}}{n!} x^n}{s \sum_{n=0}^{\infty} \frac{1}{n!}x^n} = \frac{W_0(x)}{x e^x}. $$ – Sangchul Lee Mar 28 '19 at 14:17
  • @SangchulLee Actually I was trying to integrate $ \frac {W(x)}{x} $. Since it doesn't have any elementary integral I decided to open it into a power series. Does any other simplification come in mind? –  Mar 28 '19 at 14:20
  • 2
    I guess it is much better to explain your original problem, rather than the derived problem. I suspect that you may have fallen into XY-problem. Anyway, a quick googling tells that $$ \frac{W_0(x)}{xe^x} = \sum_{n=0}^{\infty} \frac{(-1)^n}{n!} T(n)x^n, $$ where $T(n)$ counts the number of forests of rooted labeled trees using labels in a subset of ${1,\cdots, n}$, which is also the hyperbinomial transform of the constant sequence of $1$'s. (See A088957.) – Sangchul Lee Mar 28 '19 at 14:40
  • @SangchulLee I googled and didn't find anything. Could you explain what you mean by forests and trees though? –  Mar 28 '19 at 14:55
  • 1
    They are graph-theoretic terms. You may check them at this wikipedia article. In a nutshell, $T(n)$ mentioned above counts the number of certain combinatorial objects. – Sangchul Lee Mar 28 '19 at 15:30
  • @SangchulLee Thank you for your help. Have a few last questions. Does T(n) have a name as function? Could you point me to a source where the above relation is derived as I could see any explanation in A088957. –  Mar 28 '19 at 15:36
  • @SangchulLee It would be really helpful to me if you answer the above questions. In addition to these questions I have one last question, is T(n) same as the tree(n) function. If so then wont T(n) create extremely large numbers when n>2? Thanks for all your help. –  Mar 30 '19 at 18:22
  • Wow! Very interesting. – Highvoltagemath Nov 20 '19 at 03:34

0 Answers0