%I #22 Jan 25 2023 08:13:30
%S 1,4,7,5,4,29,50,-83,-185,743,1425,-5250,-9868,40530,73280,-319155,
%T -557485,2573032,4341065,-21107670,-34398290,175655925,276438452,
%U -1479202280,-2247154681,12581036223,18440253397,-107916225837,-152514334540,932452267956,1269723550920
%N a(n) = Sum_{k=0..n} (-1)^k * binomial(n+4*k+4,n-k) * Catalan(k).
%F a(n) = binomial(n+4,4) - Sum_{k=0..n-1} a(k) * a(n-k-1).
%F G.f. A(x) satisfies A(x) = 1/(1-x)^5 - x * A(x)^2.
%F G.f.: 2 / ( (1-x)^5 * (1 + sqrt( 1 + 4*x/(1-x)^5 )) ).
%F D-finite with recurrence (n+1)*a(n) +2*(-n-1)*a(n-1) +(11*n-19)*a(n-2) +20*(-n+2)*a(n-3) +15*(n-3)*a(n-4) +6*(-n+4)*a(n-5) +(n-5)*a(n-6)=0. - _R. J. Mathar_, Jan 25 2023
%o (PARI) a(n) = sum(k=0, n, (-1)^k*binomial(n+4*k+4, n-k)*binomial(2*k, k)/(k+1));
%o (PARI) my(N=40, x='x+O('x^N)); Vec(2/((1-x)^5*(1+sqrt(1+4*x/(1-x)^5))))
%Y Cf. A360058, A360059.
%Y Cf. A000108, A360051, A360057.
%K sign
%O 0,2
%A _Seiichi Manyama_, Jan 23 2023