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 A184506 G.f.: A(x) = 1 + x*G(x)/F(x) where F(x) = A(x/F(x)) and G(x) = A(x*G(x)). 3

%I #7 Mar 30 2012 18:37:25

%S 1,1,0,2,-3,27,-91,723,-3555,28338,-174027,1440582,-10280631,89422482,

%T -713833016,6548902473,-57199453969,553760916426,-5219453249126,

%U 53271930913793,-536862065044303,5767357558711960,-61733919421613462

%N G.f.: A(x) = 1 + x*G(x)/F(x) where F(x) = A(x/F(x)) and G(x) = A(x*G(x)).

%F G.f. A(x), along with F(x) = A(x/F(x)) and G(x) = A(x*G(x)), satisfy:

%F * A(x/A(x)) = 1 + x/F(x/A(x)) since G(x/A(x)) = A(x);

%F * A(x*A(x)) = 1 + x*G(x*A(x)) since F(x*A(x)) = A(x);

%F * A(x/F(x)^2) = 1 + x/[F(x)*F(x/F(x)^2)] since F(x) = G(x/F(x)^2);

%F * A(x*G(x)^2) = 1 + x * G(x)*G(x*G(x)^2) since G(x) = F(x*G(x)^2).

%e G.f.: A(x) = 1 + x + 2*x^3 - 3*x^4 + 27*x^5 - 91*x^6 + 723*x^7 -+...

%e The function F(x) = A(x/F(x)) is the g.f. of A184507 and begins:

%e F(x) = 1 + x - x^2 + 4*x^3 - 16*x^4 + 86*x^5 - 482*x^6 + 3074*x^7 - 20478*x^8 +-...

%e The function G(x) = A(x*G(x)) is the g.f. of A184508 and begins:

%e G(x) = 1 + x + x^2 + 3*x^3 + 6*x^4 + 33*x^5 + 79*x^6 + 661*x^7 + 1564*x^8 +...

%e Related expansions:

%e A(x*A(x)) = 1 + x + x^2 + 2*x^3 + 5*x^4 + 18*x^5 + 67*x^6 + 326*x^7 + 1503*x^8 +...

%e A(x/A(x)) = 1 + x - x^2 + 3*x^3 - 12*x^4 + 59*x^5 - 328*x^6 + 2021*x^7 - 13432*x^8 +-...

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

%Y Cf. A184507, A184508, A184509.

%K sign

%O 0,4

%A _Paul D. Hanna_, Jan 16 2011

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