OFFSET
1,2
COMMENTS
Compare to C(x) = C(x^3 + 3*x*C(x)^3) / C(x^2 + 2*x*C(x)^2), where C(x) = x + C(x)^2 is the g.f. of the Catalan numbers (A000108).
Conjectures:
(C1) a(n) == 1 (mod 3) iff n = 3^k for some k >= 0.
LINKS
Paul D. Hanna, Table of n, a(n) for n = 1..520
FORMULA
a(n) ~ c * d^n / n^(3/2), where d = 12.086418637032871629430806055580752... and c = 0.01774947449130389477598279659776... - Vaclav Kotesovec, Oct 10 2024
EXAMPLE
G.f.: A(x) = x + 2*x^2 + 10*x^3 + 66*x^4 + 518*x^5 + 4484*x^6 + 41424*x^7 + 399900*x^8 + 3983698*x^9 + 40622502*x^10 + 421780380*x^11 + 4442833776*x^12 + ...
where A(x) = A(x^3 + 6*x*A(x)^3) / A(x^2 + 4*x*A(x)^2).
RELATED SERIES.
A(x^2 + 4*x*A(x)^2) = x^2 + 4*x^3 + 18*x^4 + 112*x^5 + 794*x^6 + 6360*x^7 + 55266*x^8 + 509968*x^9 + 4914150*x^10 + 48889752*x^11 + 498234420*x^12 + ...
A(x^3 + 6*x*A(x)^3) = x^3 + 6*x^4 + 36*x^5 + 254*x^6 + 1980*x^7 + 16812*x^8 + 152002*x^9 + 1440828*x^10 + 14148936*x^11 + 142715046*x^12 + ...
A(x)^2 = x^2 + 4*x^3 + 24*x^4 + 172*x^5 + 1400*x^6 + 12360*x^7 + 115500*x^8 + 1123552*x^9 + 11255688*x^10 + 115291188*x^11 + 1201533048*x^12 + ...
A(x)^3 = x^3 + 6*x^4 + 42*x^5 + 326*x^6 + 2766*x^7 + 25020*x^8 + 237364*x^9 + 2332860*x^10 + 23547474*x^11 + 242620986*x^12 + ...
A(x)^2 / A(x^2 + 4*x*A(x)^2) = 1 + 6*x^2 + 36*x^3 + 354*x^4 + 3264*x^5 + 32010*x^6 + 320400*x^7 + 3276558*x^8 + 34050444*x^9 + 358651116*x^10 + 3820385664*x^11 + 41087069040*x^12 + ...
which also equals A(x)^3 / A(x^3 + 6*x*A(x)^3).
PROG
(PARI) {a(n) = my(A=[0, 1], Ax=x); for(i=1, n, A=concat(A, 0); Ax=Ser(A);
A[#A] = polcoeff( subst(Ax, x, x^3 + 6*x*Ax^3 ) - Ax*subst(Ax, x, x^2 + 4*x*Ax^2 ), #A+1)); A[n+1]}
for(n=1, 25, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Oct 04 2024
STATUS
approved