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a(n) = coefficient of x^n in A(x) where 1 = Sum_{n=-oo..+oo} (2*x + (-x)^n*A(x)^n)^n.
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%I #13 Oct 13 2023 11:09:03

%S 1,2,5,13,30,74,202,616,2126,7828,29366,110398,414214,1556848,5892713,

%T 22524354,86954484,338421674,1324660464,5204326208,20498580511,

%U 80907096678,320002290542,1268500509496,5040195484362,20073242195580,80120884387322,320442284717582,1283939790460139

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

%C 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 Paul D. Hanna, <a href="/A359673/b359673.txt">Table of n, a(n) for n = 0..400</a>

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

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

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

%F (3) 1 = Sum_{n=-oo..+oo} (-x)^(n^2) * A(x)^(n^2) / (1 - 2*(-x)^(n+1) * A(x)^n)^n.

%F (4) 1 = Sum_{n=-oo..+oo} (-x)^(n^2) * A(x)^(n^2) / (1 + 2*(-x)^(n+1) * A(x)^n)^(n+1).

%e 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 SPECIFIC VALUES.

%e A(x) = 2 at x = 0.2170550872218893465015254812376904599677836767029937...

%e A(1/5) = 1.8185729641608353079390837085677719656772552871159724...

%o (PARI) {a(n) = my(A=[1]);

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

%Y Cf. A355868.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Jan 10 2023