%I #25 May 06 2024 22:48:19

%S 1,3,4,3,5,12,6,-13,29,95,-130,-304,895,1050,-5068,-2181,27743,-5481,

%T -143532,117983,700831,-1074414,-3163138,7872784,12585117,-51587107,

%U -38040886,312988334,18178883,-1779688404,1013771196,9485832411,-11749675733,-46878057651

%N a(n) = Sum_{k=0..n} (-1)^k * binomial(n+3*k+3,n-k) * Catalan(k).

%F a(n) = binomial(n+3,3) - Sum_{k=0..n-1} a(k) * a(n-k-1).

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

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

%F D-finite with recurrence (n+1)*a(n) +(-n-2)*a(n-1) +6*(n-2)*a(n-2) +10*(-n+2)*a(n-3) +5*(n-3)*a(n-4) +(-n+4)*a(n-5)=0. - _R. J. Mathar_, Jan 25 2023

%t Table[Sum[(-1)^k Binomial[n+3k+3,n-k]CatalanNumber[k],{k,0,n}],{n,0,40}] (* _Harvey P. Dale_, May 06 2024 *)

%o (PARI) a(n) = sum(k=0, n, (-1)^k*binomial(n+3*k+3, n-k)*binomial(2*k, k)/(k+1));

%o (PARI) my(N=40, x='x+O('x^N)); Vec(2/((1-x)^4*(1+sqrt(1+4*x/(1-x)^4))))

%Y Cf. A360058, A360060.

%Y Cf. A000108, A358518, A360050.

%K sign

%O 0,2

%A _Seiichi Manyama_, Jan 23 2023