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 A023053 Number of noncrossing rooted trees with n nodes on a circle that do not have leaves at level 1. 3
 1, 0, 2, 7, 34, 171, 905, 4952, 27802, 159254, 927081, 5468960, 32621669, 196422509, 1192294778, 7288208927, 44825586130, 277196752569, 1722454028174, 10749430579118, 67346519863117, 423425225290485, 2670741276559282, 16895070479910967, 107165705513319749, 681438064187707596 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 LINKS G. C. Greubel, Table of n, a(n) for n = 0..1000 Naiomi Cameron, J. E. McLeod, Returns and Hills on Generalized Dyck Paths, Journal of Integer Sequences, Vol. 19, 2016, #16.6.1. Isaac DeJager, Madeleine Naquin, Frank Seidl, Colored Motzkin Paths of Higher Order, VERUM 2019. FORMULA a(n) = Sum_{i=0,..,n} (-1)^i*(i+1)*binomial(3*n-2*i, n-i)/(2*n-i+1). G.f.: g/(1+zg) where g = 1 + z*g^3, g(0) = 1. G.f.: g/(1+zg) where g = 2*sin(arcsin(sqrt(27*z)/2)/3)/sqrt(3*z). 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 a(n) ~ 27^(n+3/2) / (121 * sqrt(Pi) * 4^(n+1) * n^(3/2)). - Vaclav Kotesovec, Mar 17 2014 MATHEMATICA 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 *) PROG (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 (PARI) Vec((g->g/(1+x*g))(1 + serreverse(x/(1+x)^3 + O(x^25)))) \\ Andrew Howroyd, Nov 12 2017 CROSSREFS Column k=0 of A101371. Sequence in context: A241767 A222940 A227120 * A058915 A273030 A020054 Adjacent sequences:  A023050 A023051 A023052 * A023054 A023055 A023056 KEYWORD nonn AUTHOR EXTENSIONS Offset corrected by Vaclav Kotesovec, Mar 17 2014 STATUS approved

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Last modified June 21 07:27 EDT 2021. Contains 345358 sequences. (Running on oeis4.)