%I #27 Mar 11 2023 08:42:31

%S 1,2,2,2,3,3,2,4,5,0,4,13,-7,-7,48,-16,-93,180,74,-584,517,1111,-2850,

%T 207,8281,-10738,-11740,46967,-22167,-115845,211052,94468,-766989,

%U 660110,1554938,-3983408,121429,12272689,-15692006,-18841086,72792247,-31828764

%N a(n) = Sum_{k=0..n} (-1)^k * binomial(n+2*k+2,n-k) * Catalan(k).

%F a(n) = binomial(n+2,2) - Sum_{k=0..n-1} a(k) * a(n-k-1).

%F G.f. A(x) satisfies A(x) = 1/(1-x)^3 - x * A(x)^2.

%F G.f.: 2 / ( (1-x)^3 * (1 + sqrt( 1 + 4*x/(1-x)^3 )) ).

%F D-finite with recurrence (n+1)*a(n) -2*a(n-1) +2*(n-3)*a(n-2) +4*(-n+2)*a(n-3) +(n-3)*a(n-4)=0. - _R. J. Mathar_, Jan 25 2023

%o (PARI) a(n) = sum(k=0, n, (-1)^k*binomial(n+2*k+2, n-k)*binomial(2*k, k)/(k+1));

%o (PARI) my(N=50, x='x+O('x^N)); Vec(2/((1-x)^3*(1+sqrt(1+4*x/(1-x)^3))))

%Y Cf. A360059, A360060.

%Y Cf. A000108, A162481, A360049.

%K sign

%O 0,2

%A _Seiichi Manyama_, Jan 23 2023