%I #13 Nov 18 2013 12:27:24

%S 1,1,5,23,121,673,3953,23891,148321,938257,6031565,39273839,258529513,

%T 1717533025,11501168105,77546905595,526026614209,3587296445473,

%U 24580453618709,169145840067335,1168420570513177,8099287225541473,56320768883163425,392777334265548323

%N G.f. A(x) satisfies: A(x) = x + B(x)*C(x), where B(x) = x + 2*C(x)*A(x) and C(x) = x + 3*B(x)*A(x).

%H Vaclav Kotesovec, <a href="/A229811/b229811.txt">Table of n, a(n) for n = 1..300</a>

%H Vaclav Kotesovec, <a href="/A229811/a229811.txt">Recurrence (of order 9)</a>

%F G.f. A = A(x) satisfies:

%F (1) A = x + x^2*(1+2*A)*(1+3*A)/(1-6*A^2)^2.

%F (2) A = x*(1+B)/(1-3*B^2) where B = x*(1+2*A)/(1-6*A^2) is the g.f. of A229812.

%F (3) A = x*(1+C)/(1-2*C^2) where C = x*(1+3*A)/(1-6*A^2) is the g.f. of A229813.

%F The g.f.s A = A(x) (A229811), B = B(x) (A229812), C = C(x) (A229813), satisfy:

%F A*B*C = (A^2 - x*A) = (B^2 - x*B)/2 = (C^2 - x*C)/3.

%F a(n) ~ c*d^n/n^(3/2), where d = 7.438049365405038364... is the root of the equation -9 - 114*d - 442*d^2 - 792*d^3 - 660*d^4 - 432*d^5 - 192*d^6 - 24*d^7 + 8*d^8 = 0 and c = 0.0554110388537150696008515327183214457017... - _Vaclav Kotesovec_, Sep 30 2013

%e G.f.: A(x) = x + x^2 + 5*x^3 + 23*x^4 + 121*x^5 + 673*x^6 + 3953*x^7 +...

%e Related series:

%e B(x) = x + 2*x^2 + 8*x^3 + 34*x^4 + 184*x^5 + 1010*x^6 + 5936*x^7 +...

%e C(x) = x + 3*x^2 + 9*x^3 + 45*x^4 + 225*x^5 + 1275*x^6 + 7389*x^7 +...

%e where A(x) = x + B(x)*C(x).

%e (A(x)^2 - x*A(x)) = A(x)*B(x)*C(x) = x^3 + 6*x^4 + 33*x^5 + 192*x^6 + 1145*x^7 + 7038*x^8 + 44093*x^9 + 281232*x^10 + 1818513*x^11 + 11899830*x^12 +...

%o (PARI) {a(n)=local(A=x+x^2,B=x+2*x^2,C=x+3*x^2);for(i=1,n,A=x+B*C+x*O(x^n);B=x+2*A*C+x*O(x^n);C=x+3*A*B+x*O(x^n));polcoeff(A,n)}

%o for(n=1,30,print1(a(n),", "))

%o (PARI) {a(n)=local(A=x);for(i=1,n,A=x+x^2*(1+2*A)*(1+3*A)/(1-6*A^2 +x*O(x^n))^2);polcoeff(A,n)}

%o for(n=1,30,print1(a(n),", "))

%Y Cf. A229812 (B(x)), A229813 (C(x)).

%K nonn

%O 1,3

%A _Paul D. Hanna_, Sep 30 2013