A360057 - OEIS

%I #28 Mar 11 2023 08:09:11

%S 1,6,27,125,644,3643,21974,138395,898695,5970927,40386209,277127148,

%T 1924349756,13496536510,95467320600,680260392219,4878382821267,

%U 35182209381590,255000022472565,1856501085686340,13570366067586294,99554601986349471,732756800760507312

%N a(n) = Sum_{k=0..n} 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*(-5*n+1)*a(n-1) +(19*n-11)*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, binomial(n+4*k+4, n-k)*binomial(2*k, k)/(k+1));

%o (PARI) my(N=30, x='x+O('x^N)); Vec(2/((1-x)^5*(1+sqrt(1-4*x/(1-x)^5))))

%Y Cf. A086616, A162481, A358518.

%Y Cf. A000108, A360047.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Jan 23 2023