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G.f. satisfies: A(x) = 1 + x*A(x*A(x)^3).
5

%I #2 Mar 30 2012 18:37:11

%S 1,1,1,4,16,92,616,4729,40776,388057,4028230,45207583,544680014,

%T 7004865885,95694153485,1382946630490,21067128029388,337224872043659,

%U 5656357906530796,99168643108816180,1813250965008114981

%N G.f. satisfies: A(x) = 1 + x*A(x*A(x)^3).

%F G.f. satisfies: G(x) = x/[1 + A(x)*G(x)]^3 = x/A(G(x))^3 where G(x*A(x)^3) = x.

%e G.f. A(x) = 1 + x + x^2 + 4*x^3 + 16*x^4 + 92*x^5 + 616*x^6 + 4729*x^7 +...

%e A(x)^3 = 1 + 3*x + 6*x^2 + 19*x^3 + 78*x^4 + 411*x^5 + 2617*x^6 +...

%e A(x*A(x)^3) = 1 + x + 4*x^2 + 16*x^3 + 92*x^4 + 616*x^5 + 4729*x^6 +...

%e If G(x*A(x)^3) = x then

%e G(x) = x - 3*x^2 + 12*x^3 - 64*x^4 + 372*x^5 - 2385*x^6 + 15675*x^7 -+...

%e A(G(x)) = 1 + A(x)*G(x) = (x/G(x))^(1/3) where

%e A(x)*G(x) = x - 2*x^2 + 10*x^3 - 51*x^4 + 324*x^5 - 1985*x^6 + 13938*x^7 -...

%o (PARI) {a(n)=local(A=1+x+x*O(x^n));for(i=0,n,A=1+x*subst(A,x,x*A^3));polcoeff(A,n)}

%Y Cf. A143500.

%K nonn

%O 0,4

%A _Paul D. Hanna_, Aug 20 2008