A small factors sequence

[Creighton Dement]

Dear Seqfans, 

I came across a sequence given in Maple by 
seriestolist(series((1-16*x+56*x^3-140*x^4+28*x^6-16*x^7+28*x^2+56*x^5+x^8)/(x^2-x+1)^8,
x=0,50));

1, -8, -72, -120, 330, 1584, 1716, -3432, -12870, -11440, 19448, 63648,
50388, -77520, -232560, -170544, 245157, 692208, 480700, -657800,
-1776060, -1184040, 1560780, 4071600, 2629575, -3365856, -8544096,
-5379616, 6724520, 16695360, 10295472, -12620256, -30761874, -18643560,
22481940, 53956656, 32224114, -38320568, -90759240, -53524680, 62891499,
147258144, 85900584, -99884400, -231550200, -133784560, 154143080,
354201120, 202927725, -231917400

By chance, I noticed that each term of the above sequence (in the range
given) has small factors.  Pardon me for asking a vague question, but
does the above sequence illustrate anything of interest in particular?

p.s. I have visitors at the moment and, unfortunately, won't be able to
look at the above again in detail until next week (though I am quite
fortunate to have visitors!)  

Sincerely, 
Creighton

-It's a shame when the girl of your dreams would still rather be with
someone else when you're actually in a dream.