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G.f. A(x) = Sum_{n>=0} x^n * (A(x)^(n+1) + 1)^n / (1 + x*A(x)^n)^(n+1).
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%I #10 Oct 21 2019 16:14:06

%S 1,1,3,12,64,388,2547,17675,127930,957361,7363756,57974777,465801960,

%T 3811089824,31703423654,267851394004,2296630925851,19975895528930,

%U 176220976812512,1576741746108772,14312547251073466,131857909192636473,1233606830533043503,11728329063674693906,113406667874700311312,1116271813812969589106,11195131545541254173944

%N G.f. A(x) = Sum_{n>=0} x^n * (A(x)^(n+1) + 1)^n / (1 + x*A(x)^n)^(n+1).

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

%F (1) A(x) = Sum_{n>=0} x^n * (A(x)^(n+1) - 1)^n / (1 - x*A(x)^n)^(n+1).

%F (2) A(x) = Sum_{n>=0} x^n * (A(x)^(n+1) + 1)^n / (1 + x*A(x)^n)^(n+1).

%F (3) A(x) = Sum_{n>=0} x^n * Sum_{k=0..n} binomial(n,k) * (A(x)^(n+1) - A(x)^k)^(n-k).

%F (4) A(x) = Sum_{n>=0} x^n * Sum_{k=0..n} binomial(n,k) * (A(x)^(n+1) + A(x)^k)^(n-k) * (-1)^k.

%e G.f.: A(x) = 1 + x + 3*x^2 + 12*x^3 + 64*x^4 + 388*x^5 + 2547*x^6 + 17675*x^7 + 127930*x^8 + 957361*x^9 + 7363756*x^10 + 57974777*x^11 + 465801960*x^12 + ...

%e satisfies

%e A(x) = 1/(1 - x*A(x)) + x*(A(x)^2 - 1)/(1 - x*A(x))^2 + x^2*(A(x)^3 - 1)^2/(1 - x*A(x)^2)^3 + x^3*(A(x)^4 - 1)^3/(1 - x*A(x)^3)^4 + x^4*(A(x)^5 - 1)^4/(1 - x*A(x)^4)^5 + x^5*(A(x)^6 - 1)^5/(1 - x*A(x)^5)^6 + ...

%e also

%e A(x) = 1/(1 + x*A(x)) + x*(A(x)^2 + 1)/(1 + x*A(x))^2 + x^2*(A(x)^3 + 1)^2/(1 + x*A(x)^2)^3 + x^3*(A(x)^4 + 1)^3/(1 + x*A(x)^3)^4 + x^4*(A(x)^5 + 1)^4/(1 + x*A(x)^4)^5 + x^5*(A(x)^6 + 1)^5/(1 + x*A(x)^5)^6 + ...

%o (PARI) {a(n) = my(A=[1, 1]); for(i=0, n, A = concat(A, 0);

%o A[#A] = polcoeff( sum(n=0, #A+1, x^n*(Ser(A)^(n+1) + 1)^n/(1 + x*Ser(A)^n)^(n+1) ), #A-1));

%o polcoeff(Ser(A), n)}

%o for(n=0, 40, print1(a(n), ", "))

%o (PARI) {a(n) = my(A=[1, 1]); for(i=0, n, A = concat(A, 0);

%o A[#A] = polcoeff( sum(n=0, #A+1, x^n*(Ser(A)^(n+1) - 1)^n/(1 - x*Ser(A)^n)^(n+1) ), #A-1));

%o polcoeff(Ser(A), n)}

%o for(n=0, 40, print1(a(n), ", "))

%Y Cf. A324618.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Oct 19 2019