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

[israel at math.ubc.ca]

The ordinary generating function (G.f.) of a sequence a(n) is a function 
f(x) such that the Maclaurin series of f(x) is sum(n=0..infinity, 
a(n)*x^n). The exponential generating function (E.g.f.) is a function g(x) 
such that the Maclaurin series of g(x) is sum(n=0..infinity, a(n)*x^n/n!). 
See e.g. http://en.wikipedia.org/wiki/Generating_function

Cheers,
Robert


On Sep 2 2014, Wayne VanWeerthuizen 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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