Formula for New Sequence A125054 ?

[Paul D. Hanna]

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). 
 
Triangle A125053 is nice since the first column (and row sums) 
form the Secant numbers A000364 (an unexpected result!). 
  
Below I define the triangle.
 
I would be very interested in a formula, perhaps an E.g.f.? 
(I know I am asking a lot, but Seqfans surprise me all the time 
with amazing formulas). 
  
Thanks, 
      Paul 
---------------------------------------------------------
  
If we write triangle A125053 like this: 
.......................... ...1; 
.................... ...1, ...3, ...1; 
.............. ...5, ..15, ..21, ..15, ...5; 
........ ..61, .183, .285, .327, .285, .183, ..61; 
.. 1385, 4155, 6681, 8475, 9129, 8475, 6681, 4155, 1385; 

then the first nonzero term is the sum of the previous row: 
1385 = 61 + 183 + 285 + 327 + 285 + 183 + 61,

the next term is 3 times the first:
4155 = 3*1385,

and the remaining terms in each row are obtained by the rule 
illustrated by: 
6681 = 2*4155 - 1385 - 4*61 ;
8475 = 2*6681 - 4155 - 4*183 ;
9129 = 2*8475 - 6681 - 4*285 ;
8475 = 2*9129 - 8475 - 4*327 ;
6681 = 2*8475 - 9129 - 4*285 ;
4155 = 2*6681 - 8475 - 4*183 ;
1385 = 2*4155 - 6681 - 4*61 .
 
An alternate recurrence is illustrated by:
4155 = 1385 + 2*(61 + 183 + 285 + 327 + 285 + 183 + 61);
6681 = 4155 + 2*(183 + 285 + 327 + 285 + 183);
8475 = 6681 + 2*(285 + 327 + 285);
9129 = 8475 + 2*(327); 
 
and then for k>n, k<=2n, T(n,k) = T(n,2*n-k).
 
END.