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A364395 G.f. satisfies A(x) = 1 + x/A(x)*(1 + 1/A(x)^2).

%I A364395 #26 Mar 03 2024 10:32:45
%S A364395 1,2,-8,60,-552,5648,-61712,705104,-8321696,100658368,-1241281536,
%T A364395 15546987648,-197234640384,2529169695232,-32728878054144,
%U A364395 426864306146560,-5605439340018176,74050470138645504,-983432207024885760,13122261492710033408,-175836387068096147456
%N A364395 G.f. satisfies A(x) = 1 + x/A(x)*(1 + 1/A(x)^2).
%F A364395 G.f.: A(x) = 1/B(-x) where B(x) is the g.f. of A219534.
%F A364395 a(n) = (-1)^(n-1) * (1/n) * Sum_{k=0..n} binomial(n,k) * binomial(2*n+2*k-2,n-1) for n > 0.
%F A364395 D-finite with recurrence 9*n*(130549*n-619680) *(3*n-1)*(3*n-2)*a(n) +6*(-15361165*n^4 +161422948*n^3 -662268162*n^2 +955427047*n -435307620)*a(n-1) +4*(-908652649*n^4 +9061174176*n^3 -32838390812*n^2 +51018866685*n -28467674946)*a(n-2) -24*(n-3)*(50425637*n^3 -426659887*n^2 +1128823867*n -890225572)*a(n-3) -16*(n-3)*(n-4) *(4607885*n -6704077)*(2*n-9)*a(n-4)=0. - _R. J. Mathar_, Jul 25 2023
%F A364395 a(n) ~ c*(-1)^(n+1)*4^n*3F2([n-1/2, -n, n], [(n+1)/2, n/2], -1)*n^(-3/2), with c = 1/(4*sqrt(Pi)). - _Stefano Spezia_, Oct 21 2023
%F A364395 a(n) = (-1)^(n+1)*binomial(2*(n-1), n-1)*hypergeom([n-1/2, -n, n], [(n+1)/2, n/2], -1) / n. - _Peter Luschny_, Mar 03 2024
%p A364395 A364395 := proc(n)
%p A364395     if n = 0 then
%p A364395         1;
%p A364395     else
%p A364395     (-1)^(n-1)*add( binomial(n,k) * binomial(2*n+2*k-2,n-1),k=0..n)/n ;
%p A364395     end if;
%p A364395 end proc:
%p A364395 seq(A364395(n),n=0..80); # _R. J. Mathar_, Jul 25 2023
%p A364395 a := n -> `if`(n=0, 1, (-1)^(n+1)*binomial(2*(n-1), n-1)*hypergeom([n-1/2, -n, n], [(n+1)/2, n/2], -1) / n):
%p A364395 seq(simplify(a(n)), n = 0..20);  # _Peter Luschny_, Mar 03 2024
%t A364395 nmax = 20; A[_] = 1;
%t A364395 Do[A[x_] = 1 + x/A[x]*(1 + 1/A[x]^2) + O[x]^(nmax+1) // Normal, {nmax+1}];
%t A364395 CoefficientList[A[x], x] (* _Jean-François Alcover_, Mar 03 2024 *)
%o A364395 (PARI) a(n) = if(n==0, 1, (-1)^(n-1)*sum(k=0, n, binomial(n, k)*binomial(2*n+2*k-2, n-1))/n);
%Y A364395 Cf. A364393, A364397, A364399.
%Y A364395 Cf. A219534.
%K A364395 sign
%O A364395 0,2
%A A364395 _Seiichi Manyama_, Jul 22 2023