right-half-sum triangle - G.F.

[Max]

Paul,

There is another interesting sequence that is closely related to "right-half-sum" sequence.
It's


%I A096119
%S A096119 1,2,4,11,48,362
%N A096119 A096118(2^n+1).
%C A096119 Terms of A096118 which are 1 more than the previous term.
%Y A096119 Cf. A096111, A052330, A096113, A096114, A096115, A096116, A096118, A096120, A096121.
%Y A096119 Adjacent sequences: A096116 A096117 A096118 this_sequence A096120 A096121 A096122
%Y A096119 Sequence in context: A067353 A091240 A068488 this_sequence A057857 A091233 A007903
%K A096119 more,nonn,uned
%O A096119 0,2
%A A096119 Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Jul 01 2004

where

%I A096120
%S A096120 1,1,2,3,4,5,6,8,11,12,13,15,18,22,27,33,41,42,43,45,48,52,57,63,71,82,
%T A096120 94,107,120,135,153,175,202,235,236,237,239,242,246,251,257,265,276,288,
%U A096120 301
%N A096120 a(1) = 1, a(2) = 1, a(3) = a(2) +a(1), a(4) = a(3) +a(1), a(5)= a(3) +a(1) +a(2), a(6)= a(5) +a(1), a(7) = a\
   (5) +a(1) +a(2),etc. If a(2^k+1) = m then the next 2^k terms are given by a(2^k+1+r) = m + Sum {a(1) to a(r)}, (r = 1\
    to 2^k).
%C A096120 a(2^n+1) = A096119(n).
%e A096120 for k = 2, a(2^2 +1) = a(5)= 4, a(6)= a(5) +a(1)=5, a(7)= a(5)+a(1)+a(2)=6,
%e A096120 a(8) =a(5) +a(1) +a(2)+a(3)= 8,etc.
%Y A096120 Cf. A096111, A052330, A096113, A096114, A096115, A096116, A096118, A096119, A096121.
%Y A096120 Adjacent sequences: A096117 A096118 A096119 this_sequence A096121 A096122 A096123
%Y A096120 Sequence in context: A086736 A017846 A105181 this_sequence A050030 A046889 A026267
%K A096120 more,nonn,uned
%O A096120 0,3
%A A096120 Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Jul 01 2004


Current content of A096119 is incorrect. Correct sequence is

%S A096119 1 2 4 11 41 222 1761 21064 386241 11044282 500411824 36427700084 4313893200131 840463941752366 272080763704257502 147646989859379243894 135364497363933610026002
(Neil, could please fix it up?)

It would be nice to get formula, recurrence relation or g.f. for this sequence as well.

Max


Paul D Hanna wrote:
> 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
> 
>