cp's OEIS Frontend

A030980 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.

%I A030980 #22 Nov 21 2024 22:26:39
%S A030980 1,0,3,4,23,66,280,1030,4207,16852,69747,289950,1222540,5192344,
%T A030980 22239672,95864902,415730735,1812177000,7936353049,34901789568,
%U A030980 154067755503,682428824890,3032173906692,13510960371744,60360526255204,270311970889296,1213232586744900,5456560663318648
%N A030980 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 A030980 Andrew Howroyd, <a href="/A030980/b030980.txt">Table of n, a(n) for n = 1..500</a>
%H A030980 <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%F A030980 a(n) = Sum_{k=1..n} ((-1)^(n-k)*2^(n-k)*binomial(n, k)*binomial(3*k-2, k-1))/n.
%F A030980 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 A030980 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 A030980 (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
%o A030980 (PARI) seq(n)={Vec(serreverse(x/(1/(1 -x)^2 - 2*x) + O(x*x^n)))} \\ _Andrew Howroyd_, Nov 21 2024
%Y A030980 Cf. A378079.
%K A030980 nonn
%O A030980 1,3
%A A030980 _Emeric Deutsch_
%E A030980 a(25) onwards from _Andrew Howroyd_, Nov 21 2024