A138812 & A138813 Asymptotics

[Leroy Quet]

Consider these two sequences.

A138812: a(0)=1. a(n) = sum{k=0 to n-1}
floor(n/a(k)).

A138813: a(0)=1. a(n) = sum{k=0 to n-1}
ceiling(n/a(k)).

Plotting A138812 and A138813, it LOOKS like
A138812(n) ~ c*n and that A138813(n) ~ d*n, where
c and d are some constants and d > c.

But I doubt this is really the case.
I think that A138812 may actually start to rise
faster than A138813 at some point, and overtake
it. Is that hunch correct?

In any case, can someone calculate many more
terms of these sequences so as to see what
A138812 and A138813 might be asymptotical to?

Thanks,
Leroy Quet




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I don't have the book you're writing about, but it looks like you're referring
to what we call the Euler transform at the OEIS (or possibly the exponetial
transform.)

You can find information about both transforms at:
http://www.research.att.com/~njas/sequences/transforms.html

Basically, if a(n) is a sequence of the number of unlabeled connected
unlabeled structures.

If dealing with labeled structures, use exponential transform instead.

Hence A000088 unlabeled graphs is Euler transform of A001349 unlabeled
connected graphs.

A006125 labeled graphs is exponetial transform of A001187 labeled connected
graphs.

Both transforms are invertible and the inverses can be used to calculated the
connected sequence from the general sequence. The inverse exponential
transform is sometimes called the logarithmic transform.

------ Original Message ------

> Neil,
>  
> In 2005, I searched this book, looking for those enumeration formulas
involving partitions. Now after read the first pages of the chapters 1 and 4
of “Graphical Enumeration”, I continue to think that nobody published those
formulas before.
>  
> In this book often they count connected graphs of a class from results on
the total of graphs of that class. The formulas involving partitions can be
used to solve problems found in the opposite direction, i.e., to count graphs
of a class from results on the number of connected graphs of that class.
>  
> I think that I am over the shoulders of giants when I use those formulas
because they depend on results on the number of connected graphs that are
obtained with generating functions, Pólya’s theorem and group theory.
>  
> The “partition formulas” so far depending on the “real stuff “ can be
used by anyone with elementary notions of combinatorics. If “the pros” had
already found the numbers of connected graphs of a class, by a formula, or by
an OEIS sequence, etc., we can use those numbers to count “several types” of
of components and/or with restrictions on the order of their components, are
easily enumerated, both for labeled and unlabeled classes.
>  
> Washington
>  
>