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
Keywords
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
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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
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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