right-half-sum triangle - G.F.

[Max]

Ooops!

I have just noticed that A096119 is defined as

%N A096119 A096118(2^n+1).

while A096120 (incorrectly) states

%C A096120 a(2^n+1) = A096119(n).

So A096120(2^n+1) is different from A096119(n).


And what I've listed is exactly A096120(2^n+1):

1 2 4 11 41 222 1761 21064 386241 11044282 500411824 36427700084 4313893200131 840463941752366 272080763704257502 147646989859379243894 135364497363933610026002

We should create a new entry for it and refer A096120 to it (instead of A096119).

Max


Max wrote:
> 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
>>
>>
>