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%I #8 May 21 2023 00:44:04
%S 1,2,6,20,68,234,824,2956,10750,39540,146864,550096,2075432,7880046,
%T 30086704,115445028,444941028,1721720032,6686357238,26051961396,
%U 101810056296,398962013908,1567354966200,6171824148252,24355381522328,96304034538898,381506619687824
%N Expansion of g.f. A(x) satisfying 2 = Sum_{n=-oo..+oo} (-1)^n * x^n * (2*A(x) + x^(2*n-1))^(n+1).
%H Paul D. Hanna, <a href="/A363182/b363182.txt">Table of n, a(n) for n = 0..300</a>
%F G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following.
%F (1) 2 = Sum_{n=-oo..+oo} (-1)^n * x^n * (2*A(x) + x^(2*n-1))^(n+1).
%F (2) 2*x = Sum_{n=-oo..+oo} (-1)^n * x^(2*n*(n-1)) / (1 + 2*A(x)*x^(2*n+1))^(n-1).
%F (3) A(x) = 1 / Sum_{n=-oo..+oo} (-1)^n * x^n * (2*A(x) + x^(2*n-1))^n.
%F (4) A(x) = x / Sum_{n=-oo..+oo} (-1)^n * x^(3*n) * (2*A(x) + x^(2*n-1))^(n-1).
%F (5) A(x) = 1 / Sum_{n=-oo..+oo} (-1)^n * x^(2*n^2) / (1 + 2*A(x)*x^(2*n+1))^n.
%F a(n) = Sum_{k=0..floor(n/2)} A359670(n-k,n-2*k) * 2^(n-2*k) for n >= 0.
%e G.f.: A(x) = 1 + 2*x + 6*x^2 + 20*x^3 + 68*x^4 + 234*x^5 + 824*x^6 + 2956*x^7 + 10750*x^8 + 39540*x^9 + 146864*x^10 + ...
%o (PARI) {a(n) = my(A=[1]); for(i=1, n, A = concat(A, 0);
%o A[#A] = polcoeff(1 - sum(m=-#A, #A, (-1)^m * x^m * (2*Ser(A) + x^(2*m-1))^(m+1) ), #A-1)/2); A[n+1]}
%o for(n=0, 30, print1(a(n), ", "))
%Y Cf. A363142, A363183, A363184, A363185.
%Y Cf. A359670.
%K nonn
%O 0,2
%A _Paul D. Hanna_, May 20 2023
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