Sequence-Counting Sequences: Variants of A008934

[Paul D. Hanna]

Seqfans,
      Would someone wish to extend the following sequence-counting
sequences?
For positive integer values of m, they are defined by:
 
"Number of sequences (a_1, a_2,..., a_n) with a_1 = 1 
such that a_i < a_{i+1} <= m*a_i for all i."
 
For m=3: 
"Number of sequences (a_1, a_2,..., a_n) with a_1 = 1 
such that a_i < a_{i+1} <= 3*a_i for all i."
I get:
1,2,10,114,2970,182402,27392682, 
 
An example of the tree being enumerated for m=3 is: 
1
|     \
2       3
|  \ \ \     \  \  \  \  \  \ 
3,4,5,6    4,5,6,7,8,9
etc. ...
-------------------------------------------------------
For m=4: 
"Number of sequences (a_1, a_2,..., a_n) with a_1 = 1 
such that a_i < a_{i+1} <= 4*a_i for all i."
I get:
1,3,27,693,52812,12628008,

-------------------------------------------------------
For m=5: 
"Number of sequences (a_1, a_2,..., a_n) with a_1 = 1 
such that a_i < a_{i+1} <= 5*a_i for all i."
I get:
1,4,56,2704,481376,337587520,
 
-------------------------------------------------------
 
These sequences are obvious variants of A008934 (where m=2):
 
"Number of tournament sequences: sequences (a_1, a_2, ..., a_n) with
a_1 = 1 such that a_i < a_{i+1} <= 2*a_i for all i."
which begins:
1,1,2,7,41,397,6377,171886,7892642,627340987,87635138366,
21808110976027,9780286524758582,7981750158298108606,
11950197013167283686587,33046443615914736611839942
 
Reference (Cook and Kleber) for A008934:
http://www.combinatorics.org/Volume_7/PDF/v7i1r44.pdf
which supplies a formula for A008934. 
See formula in the associated table (with A008934 as column 1):
http://www.research.att.com/projects/OEIS?Anum=A093729
  
Can the formula be extended for m>2? 
Finding more terms seems very time-consuming without the use 
of some derived formula.
 
Thanks,
       Paul
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