%I #17 Jul 24 2022 12:12:12
%S 1,0,3,4,23,66,280,1030,4207,16852,69747,289950,1222540,5192344,
%T 22239672,95864902,415730735,1812177000,7936353049,34901789568,
%U 154067755503,682428824890,3032173906692,13510960371744
%N Number of planted noncrossing bushes with n nodes; i.e., rooted noncrossing trees with n nodes, root degree 1 and no nonroot nodes of degree 1.
%H <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%F a(n) = Sum_{k=1..n} ((-1)^(n-k)*2^(n-k)*binomial(n, k)*binomial(3*k-2, k-1))/n.
%F G.f.: A(z) satisfies A(z)^3 + 2z*A(z)^3 - 2A(z)^2 - 4z*A(z)^2 + A(z) + 2z*A - z = 0.
%F D-finite with recurrence -2*n*(2*n-1)*a(n) +3*n*(n-2)*a(n-1) +30*(2*n-3)*(n-2)*a(n-2) +76*(n-2)*(n-3)*a(n-3)=0. - _R. J. Mathar_, Jul 24 2022
%o (PARI) a(n) = sum(k=1, n, ((-1)^(n-k)*2^(n-k)*binomial(n, k)*binomial(3*k-2, k-1))/n) \\ _Michel Marcus_, Aug 03 2017
%K nonn
%O 1,3
%A _Emeric Deutsch_