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A363106 Expansion of g.f. A(x) satisfying 1 = Sum_{n=-oo..+oo} (-1)^n * x^(2*n) * (A(x) + x^(n-2))^(n+1).

%I A363106 #9 May 25 2023 08:58:23
%S A363106 1,2,5,14,36,98,271,752,2124,6052,17375,50292,146469,428992,1262946,
%T A363106 3734748,11089366,33048498,98819841,296388284,891436452,2688029716,
%U A363106 8124678435,24611028218,74702698749,227177047220,692084278902,2111883982538,6454350205098,19754469483978
%N A363106 Expansion of g.f. A(x) satisfying 1 = Sum_{n=-oo..+oo} (-1)^n * x^(2*n) * (A(x) + x^(n-2))^(n+1).
%H A363106 Paul D. Hanna, <a href="/A363106/b363106.txt">Table of n, a(n) for n = 0..300</a>
%F A363106 G.f. A(x) = Sum_{n>=0} a(n) * x^n may be described as follows.
%F A363106 (1) 1 = Sum_{n=-oo..+oo} (-1)^n * x^(2*n) * (A(x) + x^(n-2))^(n+1).
%F A363106 (2) 1 = Sum_{n=-oo..+oo} (-1)^n * x^(4*n) * (A(x) + x^(n-1))^n.
%F A363106 (3) x^2 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 + A(x)*x^(n+2))^(n-1).
%F A363106 (4) x^2 = Sum_{n=-oo..+oo} (-1)^(n+1) * x^(n*(n-1)) / (1 + A(x)*x^(n+2))^(n+1).
%F A363106 (5) A(x) = 1 / Sum_{n=-oo..+oo} (-1)^n * x^(2*n) * (A(x) + x^(n-2))^n.
%F A363106 (6) A(x) = 1 / Sum_{n=-oo..+oo} (-1)^n * x^(3*n-2) * (A(x) + x^(n-2))^(n-1).
%F A363106 (7) A(x) = 1 / Sum_{n=-oo..+oo} (-1)^n * x^(n^2) / (1 + A(x)*x^(n+2))^(n+1).
%F A363106 (8) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(2*n) * (A(x) + x^(n-1))^n.
%F A363106 (9) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 + A(x)*x^(n+2))^n.
%e A363106 G.f.: A(x) = 1 + 2*x + 5*x^2 + 14*x^3 + 36*x^4 + 98*x^5 + 271*x^6 + 752*x^7 + 2124*x^8 + 6052*x^9 + 17375*x^10 + 50292*x^11 + 146469*x^12 + ...
%o A363106 (PARI) {a(n) = my(A=[1], y=1); for(i=1, n, A = concat(A, 0);
%o A363106 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 A363106 for(n=0, 30, print1( a(n), ", "))
%o A363106 (PARI) {a(n) = my(A=1, y=1); for(i=1, n,
%o A363106 A = 1/sum(m=-n,n, (-1)^m * x^(2*m) * (y*A + x^(m-2) + x*O(x^n) )^m ) );
%o A363106 polcoeff( A, n, x)}
%o A363106 for(n=0, 30, print1( a(n), ", "))
%Y A363106 Cf. A359711, A363107, A363108, A363109, A363140.
%K A363106 nonn
%O A363106 0,2
%A A363106 _Paul D. Hanna_, May 24 2023