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

[Wayne VanWeerthuizen]

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