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A363109 Expansion of g.f. A(x) satisfying 4 = Sum_{n=-oo..+oo} (-1)^n * x^(2*n) * (4*A(x) + x^(n-2))^(n+1).

%I A363109 #10 May 25 2023 08:58:33
%S A363109 1,2,8,32,114,464,1840,7424,30624,126610,529832,2233584,9471888,
%T A363109 40427152,173398644,747197976,3233336302,14043404136,61203859260,
%U A363109 267565075736,1173030487248,5156102021680,22718268675276,100321210527344,443919440641296,1968097221659546
%N A363109 Expansion of g.f. A(x) satisfying 4 = Sum_{n=-oo..+oo} (-1)^n * x^(2*n) * (4*A(x) + x^(n-2))^(n+1).
%H A363109 Paul D. Hanna, <a href="/A363109/b363109.txt">Table of n, a(n) for n = 0..200</a>
%F A363109 G.f. A(x) = Sum_{n>=0} a(n) * x^n may be described as follows.
%F A363109 (1) 4 = Sum_{n=-oo..+oo} (-1)^n * x^(2*n) * (4*A(x) + x^(n-2))^(n+1).
%F A363109 (2) 4 = Sum_{n=-oo..+oo} (-1)^n * x^(4*n) * (4*A(x) + x^(n-1))^n.
%F A363109 (3) 4*x^2 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 + 4*A(x)*x^(n+2))^(n-1).
%F A363109 (4) 4*x^2 = Sum_{n=-oo..+oo} (-1)^(n+1) * x^(n*(n-1)) / (1 + 4*A(x)*x^(n+2))^(n+1).
%F A363109 (5) A(x) = 1 / Sum_{n=-oo..+oo} (-1)^n * x^(2*n) * (4*A(x) + x^(n-2))^n.
%F A363109 (6) A(x) = 1 / Sum_{n=-oo..+oo} (-1)^n * x^(3*n-2) * (4*A(x) + x^(n-2))^(n-1).
%F A363109 (7) A(x) = 1 / Sum_{n=-oo..+oo} (-1)^n * x^(n^2) / (1 + 4*A(x)*x^(n+2))^(n+1).
%F A363109 (8) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(2*n) * (4*A(x) + x^(n-1))^n.
%F A363109 (9) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 + 4*A(x)*x^(n+2))^n.
%e A363109 G.f.: A(x) = 1 + 2*x + 8*x^2 + 32*x^3 + 114*x^4 + 464*x^5 + 1840*x^6 + 7424*x^7 + 30624*x^8 + 126610*x^9 + 529832*x^10 + 2233584*x^11 + 9471888*x^12 + ...
%o A363109 (PARI) {a(n) = my(A=[1], y=4); for(i=1, n, A = concat(A, 0);
%o A363109 A[#A] = polcoeff(y - sum(n=-#A, #A, (-1)^n * x^(2*n) * (y*Ser(A) + x^(n-2))^(n+1) )/y, #A-1, x) ); A[n+1]}
%o A363109 for(n=0, 30, print1( a(n), ", "))
%o A363109 (PARI) {a(n) = my(A=1, y=4); for(i=1, n,
%o A363109 A = 1/sum(m=-n,n, (-1)^m * x^(2*m) * (y*A + x^(m-2) + x*O(x^n) )^m ) );
%o A363109 polcoeff( A, n, x)}
%o A363109 for(n=0, 30, print1( a(n), ", "))
%Y A363109 Cf. A363104, A363106, A363107, A363108.
%K A363109 nonn
%O A363109 0,2
%A A363109 _Paul D. Hanna_, May 24 2023