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A359673 a(n) = coefficient of x^n in A(x) where 1 = Sum_{n=-oo..+oo} (2*x + (-x)^n*A(x)^n)^n.

%I A359673 #13 Oct 13 2023 11:09:03
%S A359673 1,2,5,13,30,74,202,616,2126,7828,29366,110398,414214,1556848,5892713,
%T A359673 22524354,86954484,338421674,1324660464,5204326208,20498580511,
%U A359673 80907096678,320002290542,1268500509496,5040195484362,20073242195580,80120884387322,320442284717582,1283939790460139
%N A359673 a(n) = coefficient of x^n in A(x) where 1 = Sum_{n=-oo..+oo} (2*x + (-x)^n*A(x)^n)^n.
%C A359673 Given g.f. A(x), x*A(x) equals a series reversion of x*G(-x) where G(x) is the g.f. of A355868.
%H A359673 Paul D. Hanna, <a href="/A359673/b359673.txt">Table of n, a(n) for n = 0..400</a>
%F A359673 G.f. A(x) satisfies:
%F A359673 (1) 1 = Sum_{n=-oo..+oo} (2*x + (-x)^n * A(x)^n)^n.
%F A359673 (2) 1 = Sum_{n=-oo..+oo} -x^(2*n+1) * A(x)^(n+1) * (2 + (-x)^n * A(x)^(n+1))^n.
%F A359673 (3) 1 = Sum_{n=-oo..+oo} (-x)^(n^2) * A(x)^(n^2) / (1 - 2*(-x)^(n+1) * A(x)^n)^n.
%F A359673 (4) 1 = Sum_{n=-oo..+oo} (-x)^(n^2) * A(x)^(n^2) / (1 + 2*(-x)^(n+1) * A(x)^n)^(n+1).
%e A359673 G.f.: A(x) = 1 + 2*x + 5*x^2 + 13*x^3 + 30*x^4 + 74*x^5 + 202*x^6 + 616*x^7 + 2126*x^8 + 7828*x^9 + 29366*x^10 + 110398*x^11 + 414214*x^12 + ...
%e A359673 SPECIFIC VALUES.
%e A359673 A(x) = 2 at x = 0.2170550872218893465015254812376904599677836767029937...
%e A359673 A(1/5) = 1.8185729641608353079390837085677719656772552871159724...
%o A359673 (PARI) {a(n) = my(A=[1]);
%o A359673 for(i=1,n, A = concat(A,0); A[#A] = polcoeff(-1 + sum(m=-#A,#A, (2*x + (-x*Ser(A))^m)^m ), #A)/2);A[n+1]}
%o A359673 for(n=0,30,print1(a(n),", "))
%Y A359673 Cf. A355868.
%K A359673 nonn
%O A359673 0,2
%A A359673 _Paul D. Hanna_, Jan 10 2023