login
a(n) = Sum_{k=0..n} binomial(4*k,n-k) * Catalan(k).
5

%I #8 Jan 25 2023 09:08:13

%S 1,1,6,27,134,709,3892,22004,127250,749230,4476386,27071344,165398868,

%T 1019405720,6330482488,39571612357,248796862550,1572300095758,

%U 9981970108384,63633339713190,407162295120570,2614059813642256,16834457481559076

%N a(n) = Sum_{k=0..n} binomial(4*k,n-k) * Catalan(k).

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

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

%o (PARI) a(n) = sum(k=0, n, binomial(4*k, n-k)*binomial(2*k, k)/(k+1));

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

%Y Cf. A052709, A073155, A360076, A360083.

%Y Cf. A000108, A099235.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Jan 25 2023