[seqfan] Re: What does the G.f. mean in A138977 and A138978?

[Charles Greathouse]

G.f. stands for generating function (and e.g.f. for exponential generating
function). Explaining them would take a course in combinatorics, but here's
an overview:
https://en.wikipedia.org/wiki/Generating_function
and a nice (free) textbook
http://www.math.upenn.edu/~wilf/DownldGF.html

Charles Greathouse
Analyst/Programmer
Case Western Reserve University


On Tue, Sep 2, 2014 at 11:25 AM, Wayne VanWeerthuizen <waynemv at gmail.com>
wrote:

> I assume this is a very basic question and I am just missing something
> obvious to others.
>
> Regarding: A138977
> Data: 3, 19, 121, 771, 4913, 31307, 199497, 1271251, 8100769, 51620379, ...
> G.f.: (3-2*x)/(1-7*x+4*x^2). - N. J. A. Sloane, Apr 06 2008
>
> Regarding: A138978
> Data: 9, 121, 1665, 22979, 317259, 4380445, 60481881, 835088891,
> 11530288395, ...
> G.f.: -x*(8*x^2-23*x+9) / (10*x^3-31*x^2+16*x-1). [Colin Barker, Dec 03
> 2012]
>
>
> Can somebody explain those general formula to me?
>
> When I put into Sage:
>  x=3; (3-2*x)/(1-7*x+4*x^2)
> Sage gives me:
>  -3/16
> Yet, the third term of A138977 should be 121.
>
> When I put into Sage:
>  x=3; -x*(8*x^2-23*x+9) / (10*x^3-31*x^2+16*x-1)
> Sage gives me:
>  -18/19
> Yet, the third term of A138978 should be 1665.
>
>
> Am I simply not understanding how to apply these formulas?
>
> Also, if they are correct, is there a standard technique for how they were
> derived?
>
>
> By the way, I was the original author of those sequences. Is there any
> particular protocol for updating my info in the author field as that old
> email address no longer works?
>
> Also, I am thinking there should be a clarification added to those
> sequences simply saying that the matrix values are allowed to be negative.
>
> Wayne
>
>
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