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%I #30 Oct 21 2023 11:09:47
%S 1,2,-6,34,-238,1858,-15510,135490,-1223134,11320066,-106830502,
%T 1024144482,-9945711566,97634828354,-967298498358,9659274283650,
%U -97119829841854,982391779220482,-9990160542904134,102074758837531810,-1047391288012377774,10788532748880319298
%N G.f. satisfies A(x) = 1 + x/A(x)*(1 + 1/A(x)).
%F G.f.: A(x) = 1/B(-x) where B(x) is the g.f. of A027307.
%F a(n) = (-1)^(n-1) * (1/n) * Sum_{k=0..n} binomial(n,k) * binomial(2*n+k-2,n-1) = (-1)^(n-1) * A108424(n) for n > 0.
%F D-finite with recurrence n*(2*n-1)*a(n) +3*(6*n^2-10*n+3)*a(n-1) +(-46*n^2+227*n-279)*a(n-2) +2*(n-3)*(2*n-7)*a(n-3)=0. - _R. J. Mathar_, Jul 25 2023
%F a(n) ~ c*(-1)^(n-1)*4^n*2F1([-n, 2*n-1], [n], -1)*n^(-3/2), with c = 1/(4*sqrt(Pi)) = A087197/4. - _Stefano Spezia_, Oct 21 2023
%p A364394 := proc(n)
%p if n = 0 then
%p 1;
%p else
%p (-1)^(n-1)*add( binomial(n,k) * binomial(2*n+k-2,n-1),k=0..n)/n ;
%p end if;
%p end proc:
%p seq(A364394(n),n=0..80); # _R. J. Mathar_, Jul 25 2023
%o (PARI) a(n) = if(n==0, 1, (-1)^(n-1)*sum(k=0, n, binomial(n, k)*binomial(2*n+k-2, n-1))/n);
%Y Cf. A112478, A364396, A364398.
%Y Cf. A027307, A087197, A108424.
%K sign
%O 0,2
%A _Seiichi Manyama_, Jul 22 2023
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