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%I #7 Jan 21 2023 18:44:05
%S 1,1,4,9,42,187,775,3470,16085,76521,368274,1791494,8829531,43964379,
%T 220667042,1115235384,5671532510,29004157940,149056379047,
%U 769368598912,3986831368824,20733495321171,108175116519808,566067951728994,2970221822319878,15624080964153005
%N a(n) = coefficient of x^n in A(x) where x = Sum_{n=-oo..+oo} (-1)^(n-1) * x^n * (2 + x^n*A(x)^n)^n.
%H Paul D. Hanna, <a href="/A359922/b359922.txt">Table of n, a(n) for n = 0..200</a>
%F G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:
%F (1) x = Sum_{n=-oo..+oo} (-1)^(n-1) * x^n * (2 + x^n*A(x)^n)^n.
%F (2) x = Sum_{n=-oo..+oo} (-1)^(n-1) * x^(n*(n-1)) * A(x)^(n^2) / (1 + 2*x^n*A(x)^n)^n.
%e G.f.: A(x) = 1 + x + 4*x^2 + 9*x^3 + 42*x^4 + 187*x^5 + 775*x^6 + 3470*x^7 + 16085*x^8 + 76521*x^9 + 368274*x^10 + ...
%e where
%e x = ... + x^6*A(x)^9/(1 + 2*x^3*A(x)^3)^3 - x^2*A(x)^4/(1 + 2*x^2*A(x)^2)^2 + A(x)/(1 + 2*x*A(x)) - 1 + x*(2 + x*A(x)) - x^2*(2 + x^2*A(x)^2)^2 + x^3*(2 + x^3*A(x)^3)^3 + ... + (-1)^(n-1) * x^n * (2 + x^n*A(x)^n)^n + ...
%o (PARI) {a(n) = my(A=[1]); for(i=1, n, A = concat(A, 0);
%o A[#A] = polcoeff(x - sum(m=-#A, #A, (-1)^(m-1) * x^m * (2 + (x*Ser(A))^m)^m ), #A-1)); A[n+1]}
%o for(n=0, 30, print1(a(n), ", "))
%Y Cf. A359672, A359923.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Jan 18 2023
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