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G.f. satisfies A(x/(1-x)) = x*(1-sqrt(1-4*A(x)))/(2*A(x)).
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%I #10 Aug 19 2022 12:44:13

%S 1,0,1,2,5,14,39,110,323,956,2835,8648,26449,80356,253709,789110,

%T 2400681,8095578,24339611,71349218,314555855,559848480,2096495479,

%U 23012907844,-84278218251,415921423864,2974067144745

%N G.f. satisfies A(x/(1-x)) = x*(1-sqrt(1-4*A(x)))/(2*A(x)).

%F a(n) = T(n,1), T(n,m) = Sum_{k=1..n-m} T(n-m,k)*m*binomial(2*k+m-1,k+m-1)/(k+m)-binomial(n-1,k+m-2)*T(k+m-1,m), n>m, T(n,n)=1.

%o (Maxima)

%o array(BB,100,100);

%o fillarray (BB, makelist (-1, i, 1, 1000));

%o T(n,m):=if BB[n,m]=-1 then BB[n,m]:(if n=m then 1 else sum(T(n-m,k)*m*binomial(2*k+m-1,k+m-1)/(k+m)-binomial(n-1,k+m-2)*T(k+m-1,m),k,1,n-m)) else BB[n,m];

%o makelist(T(n,1),n,1,27);

%K sign

%O 1,4

%A _Vladimir Kruchinin_, Dec 04 2011