OFFSET
0,2
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..35
FORMULA
G.f. A(x) satisfies:
(1) 1 = Sum_{n>=0} ( (1+x)^n + 1/A(x) )^n / 3^(n+1).
(2) 1 = Sum_{n>=0} (1+x)^(n^2) / ( 3 - (1+x)^n/A(x) )^(n+1).
EXAMPLE
G.f.: A(x) = 1 + 5*x + 68*x^2 + 1691*x^3 + 68006*x^4 + 3845672*x^5 + 277403711*x^6 + 24067442639*x^7 + 2423632552541*x^8 + 276852951698753*x^9 + ...
such that
1 = 1/3 + ((1+x) + 1/A(x))/3^2 + ((1+x)^2 + 1/A(x))^2/3^3 + ((1+x)^3 + 1/A(x))^3/3^4 + ((1+x)^4 + 1/A(x))^4/3^5 + ((1+x)^5 + 1/A(x))^5/3^6 + ...
Also,
1 = 1/(3 - 1/A(x)) + (1+x)/(3 - (1+x)/A(x))^2 + (1+x)^4/(3 - (1+x)^2/A(x))^3 + (1+x)^9/(3 - (1+x)^3/A(x))^4 + (1+x)^16/(3 - (1+x)^4/A(x))^5 + ...
RELATED SERIES.
1/A(x) = 1 - 5*x - 43*x^2 - 1136*x^3 - 50947*x^4 - 3100946*x^5 - 234360991*x^6 - 20968984712*x^7 - 2157508424065*x^8 - 250368212385293*x^9 + ...
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Sep 21 2018
STATUS
approved