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%I #14 Nov 18 2013 12:28:25
%S 1,2,8,34,184,1010,5936,35770,221872,1401890,9005240,58596754,
%T 385519912,2560003442,17135913440,115500024010,783243325792,
%U 5340020544962,36581801008616,251678879996290,1738217149202584,12046997299005938,83759578272807440,584052716966420698
%N G.f. B(x) satisfies: B(x) = x + 2*A(x)*C(x), where A(x) = x + C(x)*B(x) and C(x) = x + 3*A(x)*B(x).
%H Vaclav Kotesovec, <a href="/A229812/b229812.txt">Table of n, a(n) for n = 1..300</a>
%H Vaclav Kotesovec, <a href="/A229812/a229812.txt">Recurrence (of order 9)</a>
%F G.f. B = B(x) satisfies:
%F (1) B = x + 2*x^2*(1+B)*(1+3*B)/(1-3*B^2)^2.
%F (2) B = x*(1+2*A)/(1-6*A^2) where A = x*(1+B)/(1-3*B^2) is the g.f. of A229811.
%F (3) B = x*(1+2*C)/(1-2*C^2) where C = x*(1+3*B)/(1-3*B^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 = 0.08214781012909829230823825161142647948... - _Vaclav Kotesovec_, Sep 30 2013
%e G.f.: B(x) = x + 2*x^2 + 8*x^3 + 34*x^4 + 184*x^5 + 1010*x^6 + 5936*x^7 +...
%e Related series:
%e A(x) = x + x^2 + 5*x^3 + 23*x^4 + 121*x^5 + 673*x^6 + 3953*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 B(x) = x + 2*A(x)*C(x).
%e (B(x)^2 - x*B(x))/2 = 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(B,n)}
%o for(n=1,30,print1(a(n),", "))
%o (PARI) {a(n)=local(B=x);for(i=1,n,B=x+2*x^2*(1+B)*(1+3*B)/(1-3*B^2 +x*O(x^n))^2);polcoeff(B,n)}
%o for(n=1,30,print1(a(n),", "))
%Y Cf. A229811 (A(x)), A229813 (C(x)).
%K nonn
%O 1,2
%A _Paul D. Hanna_, Sep 30 2013
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