Binary tree terminology, again. Re: Formula for New Sequence A125054 ?

[Antti Karttunen]

Paul D Hanna wrote:

> Seqfans,
>
>      Could someone find a formula for this new sequence?
> Sequence: A125054  begins:
>  
> 1,3,21,327,9129,396363,24615741,2068052367,225742096209,
> 31048132997523,5252064083753061,1071525520294178007,
> 259439870666594250489,73542221109962636293083,
> 24125551094579137082039181,9068240688454120376775401247,
> 3871645204706420218816959159969,
> (I can supply many more terms if needed).
>  
> The sequence forms the Central terms of a new triangle A125053
> (a variant of triangle A008301 - enumeration of binary trees).
>
Is there any more helpful description for A008301 and A125053 ?
E.g. do they relate to non-planar binary trees
(as http://www.research.att.com/~njas/sequences/A001190 )
or to (rooted, unlabeled) planar binary trees, as Catalan numbers, 
A000108, do?
Labeled or unlabeled? Rooted or non-rooted?


Cumprimentos,

Antti Karttunen
writing from Porto, Portugal.

>  
> Triangle A125053 is nice since the first column (and row sums)
> form the Euler numbers A000364 (an unexpected result!).
>  
>
> Seqfans,
>     One should always try Superseeker first!
> Superseeker says:
>
> A125054 = Binomial transform of A000182 (e.g.f. tan(x))
>
> and it holds true.
>
> I did not expect such a simple answer! 
>     Paul
>

>  
>