OFFSET
0,6
COMMENTS
Note that the continued fraction relation: 1 = F(x) - x*F(x)^k/(F(x) - x*F(x)^k/(F(x) - x*F(x)^k/(F(x) - ...))) holds when F(x) = 1 + x*F(x)^k for a fixed parameter k; this sequence explores the case where the parameter k varies over the nonnegative integers in the continued fraction expression.
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..300
FORMULA
G.f. A(x) satisfies:
(1) A(x) = P(x)/Q(x), where
P(x) = Sum_{n>=0} A(x)^(n*(n-2)) * x^n / Product_{k=1..n} (A(x)^k - 1),
Q(x) = Sum_{n>=0} A(x)^(n*(n-3)) * x^n / Product_{k=1..n} (A(x)^k - 1),
due to Ramanujan's continued fraction identity.
(2) A(x)^2 = A(x) + x*N(x)/P(x), where
N(x) = Sum_{n>=0} A(x)^(n*(n-1)) * x^n / Product_{k=1..n} (A(x)^k - 1),
P(x) = Sum_{n>=0} A(x)^(n*(n-2)) * x^n / Product_{k=1..n} (A(x)^k - 1).
(3) A(x)^2 = x/(1 - R(x)/Q(x)), where
Q(x) = Sum_{n>=0} A(x)^(n*(n-3)) * x^n / Product_{k=1..n} (A(x)^k - 1),
R(x) = Sum_{n>=0} A(x)^(n*(n-4)) * x^n / Product_{k=1..n} (A(x)^k - 1).
(4) A(x) = B(x/A(x)) where B(x) = A(x*B(x)) is the g.f. of A338748.
(5) A(x) = x/Series_Reversion( x*B(x) ) where B(x) is the g.f. of A338748.
EXAMPLE
G.f.: A(x) = 1 + x + x^3 + x^4 + 6*x^5 + 17*x^6 + 79*x^7 + 330*x^8 + 1594*x^9 + 7876*x^10 + 41433*x^11 + 226617*x^12 + ...
where
1 = A(x) - x/(A(x) - x*A(x)/(A(x) - x*A(x)^2/(A(x) - x*A(x)^3/(A(x) - x*A(x)^4/(A(x) - x*A(x)^5/(A(x) - ...)))))).
RELATED SERIES.
Given B(x) is the g.f. of A338748:
B(x) = 1 + x + x^2 + 2*x^3 + 6*x^4 + 22*x^5 + 92*x^6 + 423*x^7 + 2093*x^8 + 10994*x^9 + 60744*x^10 + 350743*x^11 + 2106422*x^12 + ...
then B(x) = A(x*B(x)) and A(x) = B(x/A(x))
where
1 = B(x) - x*B(x)/(B(x) - x*B(x)^2/(B(x) - x*B(x)^3/(B(x) - x*B(x)^4/(B(x) - x*B(x)^5/(B(x) - x*B(x)^6/(B(x) - ...)))))).
PROG
(PARI) {a(n) = my(A=[1], CF=1); for(i=1, n, A=concat(A, 0); for(i=1, #A, CF = Ser(A) - Ser(A)^(#A-i)*x/CF ); A[#A] = -polcoeff(CF, #A-1) ); H=Ser(A); A[n+1] }
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Nov 06 2020
STATUS
approved