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A386301 a(n) = 1 + Sum_{k=0..n-1} (-1)^k * a(k) * a(n-1-k).

%I A386301 #19 Jul 18 2025 08:30:00
%S A386301 1,2,1,-1,1,8,1,-28,1,134,1,-649,1,3320,1,-17497,1,94526,1,-520507,1,
%T A386301 2910896,1,-16487794,1,94393106,1,-545337199,1,3175320608,1,
%U A386301 -18615098836,1,109783526822,1,-650884962907,1,3877184797784,1,-23193307022860,1,139271612505362
%N A386301 a(n) = 1 + Sum_{k=0..n-1} (-1)^k * a(k) * a(n-1-k).
%F A386301 G.f. A(x) satisfies:
%F A386301 (1) A(x) = 1/( (1-x) * (1-x*A(-x)) ).
%F A386301 (2) A(x)*A(-x) = B(-x^2), where B(x) is the g.f. of A064641.
%F A386301 (3) A(x) = 1/(1-x) + 2*x/(1+x^2 + sqrt(1+6*x^2-3*x^4)).
%F A386301 a(2*n) = 1 and a(2*n+1) = 1 + (-1)^n * A064641(n) for n >= 0.
%o A386301 (PARI) a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=1+sum(j=0, i-1, (-1)^j*v[j+1]*v[i-j])); v;
%Y A386301 Cf. A007317, A348857, A348858, A348859.
%Y A386301 Cf. A064641, A386300.
%K A386301 sign
%O A386301 0,2
%A A386301 _Seiichi Manyama_, Jul 17 2025