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A363183 Expansion of g.f. A(x) satisfying 3 = Sum_{n=-oo..+oo} (-1)^n * x^n * (3*A(x) + x^(2*n-1))^(n+1).

%I A363183 #7 May 21 2023 00:44:08
%S A363183 1,3,11,45,193,846,3779,17169,79115,368820,1736169,8241039,39400672,
%T A363183 189567594,917146729,4459208292,21776797603,106771412718,525382657858,
%U A363183 2593665077634,12842387591191,63762186132387,317373771999035,1583380006374078,7916456438276103
%N A363183 Expansion of g.f. A(x) satisfying 3 = Sum_{n=-oo..+oo} (-1)^n * x^n * (3*A(x) + x^(2*n-1))^(n+1).
%H A363183 Paul D. Hanna, <a href="/A363183/b363183.txt">Table of n, a(n) for n = 0..300</a>
%F A363183 G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following.
%F A363183 (1) 3 = Sum_{n=-oo..+oo} (-1)^n * x^n * (3*A(x) + x^(2*n-1))^(n+1).
%F A363183 (2) 3*x = Sum_{n=-oo..+oo} (-1)^n * x^(2*n*(n-1)) / (1 + 3*A(x)*x^(2*n+1))^(n-1).
%F A363183 (3) A(x) = 1 / Sum_{n=-oo..+oo} (-1)^n * x^n * (3*A(x) + x^(2*n-1))^n.
%F A363183 (4) A(x) = x / Sum_{n=-oo..+oo} (-1)^n * x^(3*n) * (3*A(x) + x^(2*n-1))^(n-1).
%F A363183 (5) A(x) = 1 / Sum_{n=-oo..+oo} (-1)^n * x^(2*n^2) / (1 + 3*A(x)*x^(2*n+1))^n.
%F A363183 a(n) = Sum_{k=0..floor(n/2)} A359670(n-k,n-2*k) * 3^(n-2*k) for n >= 0.
%e A363183 G.f.: A(x) = 1 + 3*x + 11*x^2 + 45*x^3 + 193*x^4 + 846*x^5 + 3779*x^6 + 17169*x^7 + 79115*x^8 + 368820*x^9 + 1736169*x^10 + ...
%o A363183 (PARI) {a(n) = my(A=[1]); for(i=1, n, A = concat(A, 0);
%o A363183 A[#A] = polcoeff(1 - sum(m=-#A, #A, (-1)^m * x^m * (3*Ser(A) + x^(2*m-1))^(m+1) ), #A-1)/3); A[n+1]}
%o A363183 for(n=0, 30, print1(a(n), ", "))
%Y A363183 Cf. A363142, A363182, A363184, A363185.
%Y A363183 Cf. A359670.
%K A363183 nonn
%O A363183 0,2
%A A363183 _Paul D. Hanna_, May 20 2023