4
votes
2answers
64 views

An asymptotic term for a finite sum involving Stirling numbers

The question is a by-product at the end of this post. The following asymptotic term will ensure the convergence of some series. $$ \frac{1}{n!} \sum_{k = 1 }^{n } \frac{{n \brack k}}{k+1} = ...
2
votes
1answer
75 views

What is the family of generating functions for the *rows* of this Stirling-number matrix for whose columns they are $\exp(\exp(x)-1)-1 $?

Remark: I give much background because it might significantly help to find an idea how to generate a solution I'm analyzing properties of a certain infinite matrix $U$, for whose columns we ...
0
votes
1answer
97 views

Generating The Series

This is related to an ongoing event. It involves generating the following series : http://oeis.org/A008826 The generating Function as given in the above link is : ...
2
votes
0answers
173 views

Figuring out expression to give a integer sequence

Here given is a sequence from OEIS. The sequence is triangle of coefficients from fractional iteration of e^x - 1. Few terms are: 1, 1, 3, 1, 13, 18, 1, 50, 205, 180, 1, 201, 1865, 4245, 2700, 1, ...
8
votes
2answers
144 views

Seeking analytic proof for $\sum_{n=r}^\infty \frac{1}{n!}\left[ n-1 \atop r-1 \right] = 1$

In Blom, Holst, Sandell, "Problems and snapshots from the world of probability", section 9.4, a model of records is discussed: Elements are ordered in a sequence of increasing length according to ...