OFFSET
1,2
COMMENTS
Compare the g.f. to the following identity:
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).
FORMULA
G.f. A(x) satisfies: A( B(x) - 2*x^3/B(x) ) = x, where B(x) is described in the examples.
EXAMPLE
G.f.: A(x) = x + 2*x^2 + 8*x^3 + 38*x^4 + 196*x^5 + 1064*x^6 + 5988*x^7 + 34632*x^8 + 204672*x^9 + 1231082*x^10 + 7514052*x^11 + 46433088*x^12 +...
where A(x)^3 = A( x^3 + 6*x*A(x)^3 ).
RELATED SERIES.
A(x)^3 = x^3 + 6*x^4 + 36*x^5 + 218*x^6 + 1332*x^7 + 8208*x^8 + 50984*x^9 + 319056*x^10 + 2010528*x^11 + 12750950*x^12 + 81348948*x^13 + 521839944*x^14 + 3364421812*x^15 + 21791976192*x^16 +...
Series reversion of A(x) equals B(x) - 2*x^3/B(x) where
B(x) = x + 2*x^4 - 20*x^7 + 302*x^10 - 5436*x^13 + 108072*x^16 - 2286160*x^19 + 50475256*x^22 - 1149822240*x^25 + 26825146770*x^28 - 637700980612*x^31 + 15391872726072*x^34 - 376193675011356*x^37 + 9291840570002312*x^40 +...
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 + 6*X*A^3)^(1/3) ); polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Apr 16 2016
STATUS
approved