A023053 Number of noncrossing rooted trees with n nodes on a circle that do not have leaves at level 1.

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

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.

Taras Goy and Mark Shattuck, Determinants of Toeplitz-Hessenberg matrices with Fuss-Catalan entries, J. Integer Seq. (2026) Vol. 29, Article 26.1.2. See Eq. (18) and Corollary 9.

Index entries for sequences related to rooted trees

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

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

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: A353343 A376055 A387525 * A377963 A058915 A273030

Adjacent sequences: A023050 A023051 A023052 * A023054 A023055 A023056

Keyword

nonn

Author

Emeric Deutsch

Extensions

Offset corrected by Vaclav Kotesovec, Mar 17 2014

Status

approved