A358518 - OEIS

%I #52 Mar 11 2023 08:09:22

%S 1,5,20,85,405,2116,11766,68237,407789,2492553,15506942,97859544,

%T 624880895,4029896310,26209648212,171711104853,1132143259711,

%U 7506530891217,50019287312324,334784759816729,2249720564735567,15172573979205166,102662981205576494

%N a(n) = Sum_{k=0..n} 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) +(-9*n+2)*a(n-1) +2*(7*n-4)*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

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

%o (PARI) my(N=30, x='x+O('x^N)); Vec(2/((1-x)^4*(1+sqrt(1-4*x/(1-x)^4))))

%Y Cf. A086616, A162481, A360057.

%Y Cf. A000108, A360046.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Jan 23 2023