A376226 G.f. satisfies A(x) = A(x^3 + 6*x*A(x)^3) / A(x^2 + 4*x*A(x)^2).

1, 2, 10, 66, 518, 4484, 41424, 399900, 3983698, 40622502, 421780380, 4442833776, 47353725678, 509717438804, 5532808254500, 60492412303032, 665570138005230, 7363717939202660, 81872879608989990, 914314572022052508, 10251126194392776384, 115346231108018654736, 1302114832694059544892

Offset: 1

Author

Paul D. Hanna, Oct 04 2024

Keywords

• nonn

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.

(C2) a(n) == 2 (mod 3) iff n = A038464(k)/2 for some k >= 1, where A038464 lists the sums of 2 distinct powers of 3.

Examples

			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).
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 1..520

Programs

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),", "))

Formula

a(n) ~ c * d^n / n^(3/2), where d = 12.086418637032871629430806055580752... and c = 0.01774947449130389477598279659776... - Vaclav Kotesovec, Oct 10 2024