A360076 - OEIS

%I #28 Jan 25 2023 09:08:17

%S 1,1,5,20,90,430,2136,10937,57307,305822,1656482,9083432,50328114,

%T 281324294,1584578746,8984740485,51242962251,293772468164,

%U 1691974930584,9785378133297,56805049768157,330880419984832,1933299689139364,11328101469158554

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

%F G.f. A(x) satisfies A(x) = 1/(1 - x * (1+x)^3 * A(x)).

%F G.f.: 2 / (1 + sqrt( 1 - 4*x*(1+x)^3 )).

%o (PARI) a(n) = sum(k=0, n, binomial(3*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)^3)))

%Y Cf. A052709, A073155, A360082, A360083.

%Y Cf. A000108, A099234.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Jan 25 2023