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%I #34 Jul 26 2022 14:38:00
%S 1,0,2,7,34,171,905,4952,27802,159254,927081,5468960,32621669,
%T 196422509,1192294778,7288208927,44825586130,277196752569,
%U 1722454028174,10749430579118,67346519863117,423425225290485,2670741276559282,16895070479910967,107165705513319749,681438064187707596
%N Number of noncrossing rooted trees with n nodes on a circle that do not have leaves at level 1.
%H G. C. Greubel, <a href="/A023053/b023053.txt">Table of n, a(n) for n = 0..1000</a>
%H Naiomi Cameron, J. E. McLeod, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/McLeod/mcleod3.html">Returns and Hills on Generalized Dyck Paths</a>, Journal of Integer Sequences, Vol. 19, 2016, #16.6.1.
%H Isaac DeJager, Madeleine Naquin, Frank Seidl, <a href="https://www.valpo.edu/mathematics-statistics/files/2019/08/Drube2019.pdf">Colored Motzkin Paths of Higher Order</a>, VERUM 2019.
%H <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%F a(n) = Sum_{i=0,..,n} (-1)^i*(i+1)*binomial(3*n-2*i, n-i)/(2*n-i+1).
%F G.f.: g/(1+zg) where g = 1 + z*g^3, g(0) = 1.
%F G.f.: g/(1+zg) where g = 2*sin(arcsin(sqrt(27*z)/2)/3)/sqrt(3*z).
%F G.f.: A(x)=1/(1+x)*(1+6*x/((1+x)*G(0)-6*x)) ; G(k)= 3*x*(3*k+1)*(3*k+2) + (2*k+2)*(2*k+3) - 6*x*(k+1)*(2*k+3)*(3*k+4)*(3*k+5)/G(k+1) ; (continued fraction Euler's kind,1-step ). - _Sergei N. Gladkovskii_, Dec 29 2011
%F a(n) ~ 27^(n+3/2) / (121 * sqrt(Pi) * 4^(n+1) * n^(3/2)). - _Vaclav Kotesovec_, Mar 17 2014
%F D-finite with recurrence 2*n*(2*n+1)*a(n) +(-47*n^2+65*n-24)*a(n-1) +3*(49*n^2-167*n+148)*a(n-2) +(-65*n^2+365*n-396)*a(n-3) -12*(3*n-5)*(3*n-7)*a(n-4)=0. - _R. J. Mathar_, Jul 26 2022
%t Table[Sum[(-1)^i*(i+1)*Binomial[3*n-2*i, n-i]/(2*n-i+1), {i,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Mar 17 2014 *)
%o (PARI) for(n=0,50, print1(sum(k=0,n, (-1)^k*(k+1)*binomial(3*n-2*k,n-k)/(2*n - k+1)), ", ")) \\ _G. C. Greubel_, Feb 07 2017
%o (PARI) Vec((g->g/(1+x*g))(1 + serreverse(x/(1+x)^3 + O(x^25)))) \\ _Andrew Howroyd_, Nov 12 2017
%Y Column k=0 of A101371.
%K nonn
%O 0,3
%A _Emeric Deutsch_
%E Offset corrected by _Vaclav Kotesovec_, Mar 17 2014
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