W. Lang, Jul 13 2007

Rationals A130560/A001316 (here the denominators are conjectured to coincide with A001316).

r(n):=a(S2(3),n) with the a-sequence for the Sheffer triangle S2(3)=A035342. See the W. Lang  

link in A006232 for the notion of the a-sequence for a Sheffer matrix.

The e.g.f. for r(n) is y/finv(y) with finv(y)=(1-1/(1+y)^2)/2. Therefore, a(y)=((1+y)^2)/(1+y/2).

r(n), for n=0,..,30:

[1, 3/2, 1/2, -3/4, 3/2, -15/4, 45/4, -315/8, 315/2, -2835/4, 14175/4, -155925/8, 467775/4, -6081075/8, 
42567525/8, -638512875/16, 638512875/2, -10854718875/4, 97692469875/4, -1856156927625/8, 9280784638125/4, 
-194896477400625/8, 2143861251406875/8, -49308808782358125/16, 147926426347074375/4, -3698160658676859375/8, 
48076088562799171875/8, -1298054391195577640625/16, 9086380738369043484375/8, -263505041412702261046875/16, 
3952575621190533915703125/16]


The numerators of r(n) are, for n=0..30 are

[1, 3, 1, -3, 3, -15, 45, -315, 315, -2835, 14175, -155925, 467775, -6081075, 42567525, -638512875, 
638512875, -10854718875, 97692469875, -1856156927625, 9280784638125, -194896477400625, 2143861251406875, 
-49308808782358125, 147926426347074375, -3698160658676859375, 48076088562799171875, -1298054391195577640625, 
9086380738369043484375, -263505041412702261046875, 3952575621190533915703125]


The denominators (look like  A001316), for n=0..30:

[1, 2, 2, 4, 2, 4, 4, 8, 2, 4, 4, 8, 4, 8, 8, 16, 2, 4, 4, 8, 4, 8, 8, 16, 4, 8, 8, 16, 8, 16, 16, 32, 2, 
4, 4, 8, 4, 8, 8, 16, 4, 8, 8, 16, 8, 16, 16, 32, 4, 8, 8,...]




############################################ e.o.f. #########################################################