OFFSET
0,2
COMMENTS
The Euler transform of the power series A(x) at x=1/3 converges to the constant: c = Sum_{n>=0} (Sum_{k=0..n} C(n,k)*a(k)/3^k)/2^(n+1) = 2.080400667750319352117745232... which is the limit of S(n)^(1/3^(n-1)) where S(0)=1, S(n+1) = S(n)^3 + 1.
FORMULA
G.f. begins: A(x) = (1+m*x) + m^m*x^(m+1)/(1+m*x)^(m-1) + ... at m=3.
EXAMPLE
The iteration begins:
F(0) = 1,
F(1) = 1 + 3*x,
F(2) = 1 + 9*x + 27*x^2 + 27*x^3 + 81*x^4,
F(3) = 1 + 27*x + 324*x^2 + 2268*x^3 + 10449*x^4 + ... + 1594323*x^13.
The 3^(n-1)-th roots of F(n) tend to the limit of A(x):
F(1)^(1/3^0) = 1 + 3*x
F(2)^(1/3^1) = 1 + 3*x + 27*x^4 - 162*x^5 + 729*x^6 - 2916*x^7 + ...
F(3)^(1/3^2) = 1 + 3*x + 27*x^4 - 162*x^5 + 729*x^6 - 2916*x^7 + ...
PROG
(PARI) {a(n)=local(F=1, A, L); if(n==0, A=1, L=ceil(log(n+1)/log(3)); for(k=1, L, F=F^3+(3*x)^((3^k-1)/2)); A=polcoeff((F+x*O(x^n))^(1/3^(L-1)), n)); A}
CROSSREFS
KEYWORD
sign
AUTHOR
Paul D. Hanna, Dec 07 2004
STATUS
approved