Number of Self-Convolution Sequences of Length n

[Paul D. Hanna]

Dear Seqfans,
    The following question has come to mind.
"How many sequences of length n are formed from the 
self-convolution of another integer sequence?"
Without restrictions, this question is indeterminate.
To make the question interesting, one needs to include 
natural conditions, such as given by the sequence: 
 
http://www.research.att.com/projects/OEIS?Anum=A008934
1,1,2,7,41,397,6377,171886,7892642,627340987,87635138366,
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.
 
Below I give a few cases that I find interesting.
 
Would someone like to compute one of these sequences?
 
Thanks,
   Paul 
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SEQUENCE (1) - square sequences with growth factor 2:

"Number of increasing sequences of length n starting with s(1)=1 
such that the initial n terms, (s(1),s(2),s(3),...,s(n)), satisfy 
s(i+1) <= 2*s(i) for 1<=i<n, and form the initial n terms of the 
self-convolution square of some integer sequence."
 
I calculate an upper bound for the initial terms to be:
{1,1,1,2,5,22,142,1583,28700,909520,...}.
 
EXAMPLE.
Starting with 1, only s(2)=2 is the next valid number; sequence [1,2] 
forms the initial terms of a self-convolution of an integer sequence. 
A sequence beginning [1,2,s(3),...] can be a self-convolution 
and satisfy s(3) <= 2*2 only at s(3) = 3; giving 1 valid sequence. 
A sequence beginning [1,2,3,a_4,...] can be a self-convolution 
and satisfy s(4) <= 2*3 only at s(4) = 4 or 6; giving 2 valid sequences.
Likewise, for the next term s(5) there are 5 valid sequences: 
[1,2,3,4,5],[1,2,3,4,7],[1,2,3,6,7],[1,2,3,6,9],[1,2,3,6,11]. 
 
So the SEQUENCE (1) begins: {1,1,1,2,5,...};
what is the rest of the sequence?
 
Also of interest is the sum of the nodes at each 
generation in the tree of allowed sequence terms, 
such as 1=1, 2=2, 3=3, 10=4+6, 39=5+7+7+9+11, etc. 
This forms a sequence with an upper bound on initial terms being: 
{1,2,3,10,39,284,3024,57400,1790340,...}. 
 
 
SEQUENCE (2) - square sequences with growth factor 3:
 
"Number of increasing sequences of length n starting with s(1)=1 
such that the initial n terms, (s(1),s(2),s(3),...,s(n)), satisfy 
s(i+1) <= 3*s(i) for 1<=i<n, and form the initial n terms of the 
self-convolution square of some integer sequence."
 
SEQUENCE (3) - cube sequences with growth factor 3:
 
"Number of increasing sequences of length n starting with s(1)=1 
such that the initial n terms, (s(1),s(2),s(3),...,s(n)), satisfy 
s(i+1) <= 3*s(i) for 1<=i<n, and form the initial n terms of the 
self-convolution cube of some integer sequence."
 
Other obvious variants may also be worth computing.