Let us define a superincreasing sequence of natural numbers $b_1, b_2, \ldots,b_n$:
$ b_{i+1} > \sum_{j=1}^i b_j $
or, as we talk about natural numbers:
$ b_{i+1} \geq 1+ \sum_{j=1}^i b_j. $
What is the number of such sequences with the following property:
$ \sum_{j=1}^n b_j \leq 2^{n+1}-1. $
A simple lower-bound estimate is $n+1$ - this is the number of superincreasing sequences which consist of powers of 2, i.e. $n$-combinations of set $\{1,2,4,\ldots,2^n\}$.
In case this is complicated to find simple exact formula, estimates will be interesting too.
UPD 1. Program generation showed the following results for the first values of $n$: 3, 9, 35, 201, 1827, 27337, 692003, 30251721... This is OEIS A125792.
UPD 2. It seems the number I was looking for is discussed in the paper V. Bakoev, Algorithmic approach to counting of certain types $m$-ary partitions (also available here).