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A384270 G.f. satisfies A(x) = A(x^4 + 4*x*A(x)^4) / A(x^3 + 3*x*A(x)^3).

%I A384270 #12 Jul 16 2025 03:21:27
%S A384270 1,1,4,21,130,888,6408,48063,371020,2927770,23510106,191487656,
%T A384270 1578141446,13136158810,110276181794,932581374945,7937416827088,
%U A384270 67940240601388,584459497206176,5050440687673800,43818234017497418,381559984178906078,3333557306752862686,29212187993468834734,256697187873464085446
%N A384270 G.f. satisfies A(x) = A(x^4 + 4*x*A(x)^4) / A(x^3 + 3*x*A(x)^3).
%C A384270 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 A384270 Conjecture: a(n) is odd iff n = 2^k for k >= 0.
%H A384270 Paul D. Hanna, <a href="/A384270/b384270.txt">Table of n, a(n) for n = 1..520</a>
%F A384270 a(n) ~ c * d^n / n^(3/2), where d = 9.35905430294846226886359252511409062627992765028... and c = 0.016106850486841401468225701097845183576331749... - _Vaclav Kotesovec_, Jul 16 2025
%e A384270 G.f.: A(x) = x + x^2 + 4*x^3 + 21*x^4 + 130*x^5 + 888*x^6 + 6408*x^7 + 48063*x^8 + 371020*x^9 + 2927770*x^10 + 23510106*x^11 + 191487656*x^12 + ...
%e A384270 where A(x) = A(x^4 + 4*x*A(x)^4) / A(x^3 + 3*x*A(x)^3).
%e A384270 RELATED SERIES.
%e A384270 A(x^4 + 4*x*A(x)^4) = x^4 + 4*x^5 + 16*x^6 + 88*x^7 + 545*x^8 + 3676*x^9 + 26368*x^10 + 197048*x^11 + 1517156*x^12 + ...
%e A384270 A(x^3 + 3*x*A(x)^3) = x^3 + 3*x^4 + 9*x^5 + 46*x^6 + 270*x^7 + 1755*x^8 + 12325*x^9 + 90774*x^10 + 691398*x^11 + 5401614*x^12 + ...
%e A384270 A(x)^3 / A(x^3 + 3*x*A(x)^3) = 1 + 6*x^2 + 24*x^3 + 180*x^4 + 1272*x^5 + 9468*x^6 + 72528*x^7 + 568890*x^8 + 4544616*x^9 + 36854304*x^10 + 302604984*x^11 + 2510760870*x^12 + ...
%e A384270 which also equals A(x)^4 / A(x^4 + 4*x*A(x)^4).
%o A384270 (PARI) {a(n) = my(A=[0, 1], Ax=x); for(i=1, n, A=concat(A, 0); Ax=Ser(A);
%o A384270 A[#A] = polcoeff( subst(Ax, x, x^4 + 4*x*Ax^4 ) - Ax*subst(Ax, x, x^3 + 3*x*Ax^3 ), #A+2)); A[n+1]}
%o A384270 for(n=1, 25, print1(a(n), ", "))
%Y A384270 Cf. A376226.
%K A384270 nonn
%O A384270 1,3
%A A384270 _Paul D. Hanna_, Jul 10 2025