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A364393 G.f. satisfies A(x) = 1 + x*(1 + 1/A(x)^2).

%I A364393 #24 Oct 21 2023 11:10:12
%S A364393 1,2,-4,20,-120,800,-5696,42416,-326304,2572992,-20685696,168920704,
%T A364393 -1397257472,11682707712,-98578346496,838369268480,-7178912946688,
%U A364393 61842549386240,-535575159363584,4660216874719232,-40722264390799360,357204260381327360
%N A364393 G.f. satisfies A(x) = 1 + x*(1 + 1/A(x)^2).
%F A364393 G.f.: A(x) = 1/B(-x) where B(x) is the g.f. of A346626.
%F A364393 a(n) = (-1)^(n-1) * (1/n) * Sum_{k=0..n} binomial(n,k) * binomial(n+2*k-2,n-1) for n > 0.
%F A364393 D-finite with recurrence 2*n*(2*n-1)*a(n) +(35*n^2-64*n+24) *a(n-1) +(-35*n^2+205*n-288) *a(n-2) +2*(-43*n^2+341*n-660) *a(n-3) -4*(7*n-30)*(n-5) *a(n-4) -8*(n-5)*(n-6)*a(n-5)=0. - _R. J. Mathar_, Jul 25 2023
%F A364393 a(n) = (-1)^(n-1)*n*3F2([1-n, (n+1)/2, n/2+1], [3/2, 2], -1) for n > 1. - _Stefano Spezia_, Oct 21 2023
%p A364393 A364393 := proc(n)
%p A364393     if n = 0 then
%p A364393         1;
%p A364393     else
%p A364393         (-1)^(n-1)*add( binomial(n,k) * binomial(n+2*k-2,n-1),k=0..n)/n ;
%p A364393     end if;
%p A364393 end proc:
%p A364393 seq(A364393(n),n=0..70); # _R. J. Mathar_, Jul 25 2023
%t A364393 m = 22;
%t A364393 A[_] = 1;
%t A364393 Do[A[x_] = 1 + x*(1 + 1/A[x]^2) + O[x]^m // Normal, {m}];
%t A364393 CoefficientList[A[x], x] (* _Jean-François Alcover_, Sep 05 2023 *)
%o A364393 (PARI) a(n) = if(n==0, 1, (-1)^(n-1)*sum(k=0, n, binomial(n, k)*binomial(n+2*k-2, n-1))/n);
%Y A364393 Cf. A364395, A364397, A364399.
%Y A364393 Cf. A346626.
%K A364393 sign
%O A364393 0,2
%A A364393 _Seiichi Manyama_, Jul 22 2023