%I #10 May 10 2023 09:42:18

%S 1,1,6,33,198,1204,7522,48270,316281,2110018,14293494,98054885,

%T 679735489,4753912524,33504984427,237767467381,1697719206178,

%U 12188097989345,87913304459342,636736565338008,4628839922257617,33767007201285762,247145222148251103,1814452818239003585

%N Expansion of g.f. A(x) satisfying x*A(x) = Sum_{n=-oo..+oo} (-1)^n * x^n * A(x)^n * (A(x)^n + x^n)^n.

%C First negative term is a(51) = -47152346702575235627205086026135269902810693.

%H Paul D. Hanna, <a href="/A361776/b361776.txt">Table of n, a(n) for n = 0..200</a>

%F G.f. A(x) = Sum_{n>=0} a(n)*x^n may be defined by the following.

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

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

%e G.f.: A(x) = 1 + x + 6*x^2 + 33*x^3 + 198*x^4 + 1204*x^5 + 7522*x^6 + 48270*x^7 + 316281*x^8 + 2110018*x^9 + 14293494*x^10 + ...

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

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

%o for(n=0,30,print1(a(n),", "))

%Y Cf. A361775.

%K sign

%O 0,3

%A _Paul D. Hanna_, May 08 2023