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A363108 Expansion of g.f. A(x) satisfying 3 = Sum_{n=-oo..+oo} (-1)^n * x^(2*n) * (3*A(x) + x^(n-2))^(n+1).

%I A363108 #10 May 25 2023 08:58:29
%S A363108 1,2,7,26,86,318,1165,4312,16318,62020,238165,921980,3590145,14067188,
%T A363108 55399442,219172028,870736366,3472155062,13892694747,55759406580,
%U A363108 224427809830,905659181212,3663475842865,14851965523630,60334690089827,245572722474460,1001306332164918
%N A363108 Expansion of g.f. A(x) satisfying 3 = Sum_{n=-oo..+oo} (-1)^n * x^(2*n) * (3*A(x) + x^(n-2))^(n+1).
%H A363108 Paul D. Hanna, <a href="/A363108/b363108.txt">Table of n, a(n) for n = 0..300</a>
%F A363108 G.f. A(x) = Sum_{n>=0} a(n) * x^n may be described as follows.
%F A363108 (1) 3 = Sum_{n=-oo..+oo} (-1)^n * x^(2*n) * (3*A(x) + x^(n-2))^(n+1).
%F A363108 (2) 3 = Sum_{n=-oo..+oo} (-1)^n * x^(4*n) * (3*A(x) + x^(n-1))^n.
%F A363108 (3) 3*x^2 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 + 3*A(x)*x^(n+2))^(n-1).
%F A363108 (4) 3*x^2 = Sum_{n=-oo..+oo} (-1)^(n+1) * x^(n*(n-1)) / (1 + 3*A(x)*x^(n+2))^(n+1).
%F A363108 (5) A(x) = 1 / Sum_{n=-oo..+oo} (-1)^n * x^(2*n) * (3*A(x) + x^(n-2))^n.
%F A363108 (6) A(x) = 1 / Sum_{n=-oo..+oo} (-1)^n * x^(3*n-2) * (3*A(x) + x^(n-2))^(n-1).
%F A363108 (7) A(x) = 1 / Sum_{n=-oo..+oo} (-1)^n * x^(n^2) / (1 + 3*A(x)*x^(n+2))^(n+1).
%F A363108 (8) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(2*n) * (3*A(x) + x^(n-1))^n.
%F A363108 (9) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 + 3*A(x)*x^(n+2))^n.
%e A363108 G.f.: A(x) = 1 + 2*x + 7*x^2 + 26*x^3 + 86*x^4 + 318*x^5 + 1165*x^6 + 4312*x^7 + 16318*x^8 + 62020*x^9 + 238165*x^10 + 921980*x^11 + 3590145*x^12 + ...
%o A363108 (PARI) {a(n) = my(A=[1], y=3); for(i=1, n, A = concat(A, 0);
%o A363108 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 A363108 for(n=0, 30, print1( a(n), ", "))
%o A363108 (PARI) {a(n) = my(A=1, y=3); for(i=1, n,
%o A363108 A = 1/sum(m=-n,n, (-1)^m * x^(2*m) * (y*A + x^(m-2) + x*O(x^n) )^m ) );
%o A363108 polcoeff( A, n, x)}
%o A363108 for(n=0, 30, print1( a(n), ", "))
%Y A363108 Cf. A359713, A363106, A363107, A363109.
%K A363108 nonn
%O A363108 0,2
%A A363108 _Paul D. Hanna_, May 24 2023