|
%I #25 Nov 13 2017 03:12:20
%S 0,1,1,7,18,80,284,1169,4628,19137,79165,333058,1410608,6029816,
%T 25941384,112315945,488862888,2138161043,9391903131,41414729419,
%U 183264846010,813564012660,3622193670040,16170171489820,72364908958800,324586284275500,1458976377988636
%N Number of noncrossing nonplanted bushes with n nodes, i.e., rooted noncrossing trees with n nodes and no nodes of degree 1.
%H Vincenzo Librandi, <a href="/A030982/b030982.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%F a(n) = 5*Sum_{k=2..n} ((-1)^(n-k)*2^(n-k)*k*C(n,k)*C(3*k-2,k-2)/(2*k+1))/n.
%F Recurrence: 2*n*(2*n+1)*a(n) = (n-1)*(11*n-12)*a(n-1) + 6*(9*n^2-21*n+8) * a(n-2) - 4*(n-3)*(11*n-56)*a(n-3) - 152*(n-4)*(n-3)*a(n-4). - _Vaclav Kotesovec_, Oct 24 2012
%F a(n) ~ 5*19^(n+1/2)/(27*sqrt(Pi)*4^(n+1)*n^(3/2)). - _Vaclav Kotesovec_, Oct 24 2012
%F a(n) = A030981(n) - A030980(n). - _Andrew Howroyd_, Nov 12 2017
%t Table[5*Sum[(-1)^(n-k)*2^(n-k)*k*Binomial[n,k]*Binomial[3*k-2,k-2]/ (2*k+1),{k,2,n}]/n,{n,1,20}] (* _Vaclav Kotesovec_, Oct 24 2012 *)
%o (PARI) a(n) = 5*sum(k=2, n, (-1)^(n-k)*2^(n-k)*k*binomial(n,k)*binomial(3*k-2,k-2)/(2*k+1))/n; \\ _Andrew Howroyd_, Nov 12 2017
%Y Cf. A030980, A030981.
%K nonn,easy
%O 1,4
%A _Emeric Deutsch_
|
