OFFSET
1,2
COMMENTS
Compare the g.f. to the following identities:
(1) C(x) = C( x^2 + 2*x*C(x)^2 )^(1/2),
(2) C(x) = C( x^3 + 3*x*C(x)^3 )^(1/3),
where C(x) = x + C(x)^2 is a g.f. of the Catalan numbers (A000108).
LINKS
Paul D. Hanna, Table of n, a(n) for n = 1..500
FORMULA
G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
(1.a) A(x)^3 = A( x*A(x)^2 * (1 + 3*A(x)) ).
(1.b) A(x)^9 = A( x*A(x)^8 * (1 + 3*A(x))*(1 + 3*A(x)^3) ).
(1.c) A(x)^27 = A( x*A(x)^26 * (1 + 3*A(x))*(1 + 3*A(x)^3)*(1 + 3*A(x)^9) ).
(1.d) A(x)^(3^n) = A( x*A(x)^(3^n-1) * Product_{k=0..n-1} (1 + 3*A(x)^(3^k)) ).
(2) A(x) = x * Product_{n>=0} (1 + 3*A(x)^(3^n)).
(3) A(x) = Series_Reversion( x / Product_{n>=0} (1 + 3*x^(3^n)) ).
(4) A(x) = x * Sum_{n>=0} A117940(n) * A(x)^n, where g.f. of A117940 equals Product{k>=0} 1 + 3*x^(3^k).
a(n) ~ c * d^n / n^(3/2), where d = 6.5583689184153129045048... and c = 0.129061736750222730297... - Vaclav Kotesovec, Apr 05 2024
The radius of convergence r of g.f. A(x) and A(r) satisfy 1 = Sum_{n>=0} 3^(n+1) * A(r)^(3^n) / (1 + 3*A(r)^(3^n)) and r = A(r) / Product_{n>=0} (1 + 3*A(r)^(3^n)), where r = 0.1524769363297159918479... = 1/d (d is given above) and A(r) = 0.3905308673397427979651361312666180120359942797557... - Paul D. Hanna, May 22 2024
EXAMPLE
G.f.: A(x) = x + 3*x^2 + 9*x^3 + 30*x^4 + 126*x^5 + 648*x^6 + 3591*x^7 + 19953*x^8 + 110079*x^9 + 610500*x^10 + 3440493*x^11 + 19742616*x^12 + ...
where A(x)^3 = A( x*A(x)^2 + 3*x*A(x)^3 ).
PROG
(PARI) {a(n) = my(A=[0, 1]); for(i=1, n, A=concat(A, 0);
F=Ser(A); A[#A] = polcoeff(subst(F, x, x*F^2 + 3*x*F^3) - F^3, #A+1) ); A[n+1]}
for(n=1, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Mar 27 2024
STATUS
approved