Rationals  A120084(n)/A120085(n):

Expansion of generalized Debye function D(2,x) for |x|<2*pi.

D(2,x):= (2/x^2)*int(t^2/(exp(t)-1),t,0,x). 


r(n):=[x^n](1 - 2*x/2*(2+1) + sum((B(2*k)/((k+1)*(2*k)!))*x^(2*k),0,..infinity)) (in lowest terms), |x|<2*pi,

with the Bernoulli numbers B(2*k):=A00367(k)/A002445(k).

for n=0,,40:

[1, -1/3, 1/24, 0, (-1)/2160, 0, 1/120960, 0, -1/6048000, 0, 1/287400960, 0, -691/9153720576000, 0, 
1/597793996800, 0, -3617/96035605585920000, 0, 43867/51090942171709440000, 0, 
-174611/8831434289681203200000, 0, 77683/169213200472701665280000, 0, -236364091/22019713777512667702886400000, 
0, 657931/2605883287279605645312000000, 0, -3392780147/568398985311837982792089600000000, 0, 
1723168255201/12156644666336641093725472161792000000, 0, -7709321041217/2281344356215122906549780165427200000000,
 0, 151628697551/1875596605663367483031929990676480000000, 0, 
-26315271553053477373/13564591583996299833495787357035426742272000000000, 0, 
154210205991661/3303300741894322811115835099579416576000000000, 0, 
-261082718496449122051/231826009429225183259618530030344409039503360000000000]


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