right-half-sum triangle - G.F.

[Paul D Hanna]

Dear Seqfans,
     Here is a nice formal g.f. for Max Alekseyev's A107354:
 
G.f.: 1 = Sum_{n>=0} a(n)*x^n*(1-x)^(2^n)
 
e.g.,
 
1 = 1*(1-x) + 1*x*(1-x)^2 + 2*x^2*(1-x)^4 + 7*x^3*(1-x)^8 +44*x^4*(1-x)^16 + 516*x^5*(1-x)^32 + 11622*x^6*(1-x)^64 +...
   
Matrix representation of g.f.
-----------------------------
Define a triangular matrix T where:

T(n,k) = [x^(n-k)] (1-x)^(2^k)
or, in PARI: 
T(n,k)=polcoeff((1-x)^(2^k),n-k)

Matrix T begins:
1;
-1,1;
0,-2,1;
0,1,-4,1;
0,0,6,-8,1;
0,0,-4,28,-16,1;
0,0,1,-56,120,-32,1;
0,0,0,70,-560,496,-64,1;
0,0,0,-56,1820,-4960,2016,-128,1;
0,0,0,28,-4368,35960,-41664,8128,-256,1;
...
Matrix inverse, T^-1, begins:
1;
1,1;
2,2,1;
7,7,4,1;
44,44,26,8,1;
516,516,308,100,16,1;
11622,11622,6959,2296,392,32,1;
512022,512022,306888,101754,17712,1552,64,1;
44588536,44588536,26732904,8877272,1554404,139104,6176,128,1;
...
where column 0 is A107354.

There are several more properties of A107354 
that I have found, but this is enough for now.
Paul