A376226 - OEIS

%I #9 Oct 10 2024 11:46:25

%S 1,2,10,66,518,4484,41424,399900,3983698,40622502,421780380,

%T 4442833776,47353725678,509717438804,5532808254500,60492412303032,

%U 665570138005230,7363717939202660,81872879608989990,914314572022052508,10251126194392776384,115346231108018654736,1302114832694059544892

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

%C 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).

%C Conjectures:

%C (C1) a(n) == 1 (mod 3) iff n = 3^k for some k >= 0.

%C (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.

%H Paul D. Hanna, <a href="/A376226/b376226.txt">Table of n, a(n) for n = 1..520</a>

%F a(n) ~ c * d^n / n^(3/2), where d = 12.086418637032871629430806055580752... and c = 0.01774947449130389477598279659776... - _Vaclav Kotesovec_, Oct 10 2024

%e 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 + ...

%e where A(x) = A(x^3 + 6*x*A(x)^3) / A(x^2 + 4*x*A(x)^2).

%e RELATED SERIES.

%e 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 + ...

%e 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 + ...

%e 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 + ...

%e 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 + ...

%e 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 + ...

%e which also equals A(x)^3 / A(x^3 + 6*x*A(x)^3).

%o (PARI) {a(n) = my(A=[0,1],Ax=x); for(i=1,n, A=concat(A,0); Ax=Ser(A);

%o 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]}

%o for(n=1,25,print1(a(n),", "))

%K nonn

%O 1,2

%A _Paul D. Hanna_, Oct 04 2024