OFFSET
1,2
COMMENTS
Compare to 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..401
FORMULA
G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
(1) A(x)^3 = A( x^3 + 12*x*A(x)^3 ).
(2) A(x)^9 = A( x^9 + 36*x^7*A(x)^3 + 432*x^5*A(x)^6 + 1740*x^3*A(x)^9 + 144*x*A(x)^12 ).
(3) R(x)^3 = R(x^3) - 12*x^3*R(x) where R(x) is the series reversion of A(x).
(4) A( B(x) - 4*x^3/B(x) ) = x where B(x) is described in the examples.
a(n) ~ c * d^n / n^(3/2), where d = 13.104641971... and c = 0.10266771... - Vaclav Kotesovec, Jan 27 2026
EXAMPLE
G.f.: A(x) = x + 4*x^2 + 32*x^3 + 300*x^4 + 3024*x^5 + 31808*x^6 + 344272*x^7 + 3804736*x^8 + 42731520*x^9 + 486196388*x^10 + ...
where A(x)^3 = A( x^3 + 12*x*A(x)^3 ).
RELATED SERIES.
A(x)^3 = x^3 + 12*x^4 + 144*x^5 + 1732*x^6 + 20880*x^7 + 252288*x^8 + 3055136*x^9 + 37077888*x^10 + ...
Series reversion of A(x) equals R(x) = B(x) - 4*x^3/B(x) where
B(x) = x + 20*x^4 - 1680*x^7 + 223260*x^10 - 35677680*x^13 + 6300518720*x^16 - 1184728754240*x^19 + 232591757097920*x^22 - 47123699986542080*x^25 + ...
where B(x) = (R(x) + sqrt(R(x)^2 + 16*x^3))/2 is a series trisection of R(x) with R(A(x)) = x.
The series reversion of A(x) begins
R(x) = x - 4*x^2 + 20*x^4 + 80*x^5 - 1680*x^7 - 8320*x^8 + 223260*x^10 + 1193840*x^11 - 35677680*x^13 - 198425920*x^14 + ...
where R(x)^3 = R(x^3) - 12*x^3*R(x).
SPECIFIC VALUES.
A(t) = 1/5 at t = 0.0760803602365477509640767946761615820467284570636924...
A(t) = 1/6 at t = 0.0743825519512132015029644796035774538260279235312880...
A(t) = 1/7 at t = 0.0717872014352652118492063869289246553804123373403773...
A(t) = 1/8 at t = 0.0687921839160880702120477745249325625782658063203404...
A(t) = 1/9 at t = 0.0656707734153371845897720241327365679578198722259356...
A(t) = 1/10 at t = 0.0625783449270176468015112698101223059381555236181601...
A(1/14) = 0.140357805905579577459080496546205788896865457949170...
A(1/15) = 0.115184030758394387577917540300038966424602075703692...
A(1/16) = 0.099743258982035577782399133907101902639596116802265...
PROG
(PARI) {a(n) = my(A=x+x^2, X=x+x*O(x^n)); for(i=1, n, A = subst(A, x, x^3 + 12*X*A^3)^(1/3) ); polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jan 15 2026
STATUS
approved
