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

Author

Emeric Deutsch

Keywords

• nonn

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.
• Index entries for sequences related to rooted trees

Crossrefs

Column k=0 of A101371.

Programs

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 *)

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

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

Extensions

Offset corrected by Vaclav Kotesovec, Mar 17 2014