OFFSET
0,2
COMMENTS
Sum of root degrees of all noncrossing trees on nodes on a circle. - Emeric Deutsch
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..200
Paul Drube, Raised k-Dyck paths, arXiv:2206.01194 [math.CO], 2022. See Appendix pp. 14-15.
H. M. Finucan, Some decompositions of generalized Catalan numbers, pp. 275-293 of Combinatorial Mathematics IX. Proc. Ninth Australian Conference (Brisbane, August 1981). Ed. E. J. Billington, S. Oates-Williams and A. P. Street. Lecture Notes Math., 952, Springer-Verlag, 1982.
Emanuele Munarini, Shifting Property for Riordan, Sheffer and Connection Constants Matrices, Journal of Integer Sequences, Vol. 20 (2017), Article 17.8.2.
Joris Nieuwveld, Fractions, Functions and Folding. A Novel Link between Continued Fractions, Mahler Functions and Paper Folding, Master's Thesis, arXiv:2108.11382 [math.NT], 2021.
Chin Hee Pah, Single polygon counting on Cayley Tree of order 3, J. Stat. Phys. 140 (2010), 198-207.
FORMULA
a(n) = 2*binomial(3*n+3,n)/(n+2). - Emeric Deutsch
a(n) = (n+1) * A000139(n+1). - F. Chapoton, Feb 23 2024
G.f.: hypergeom( [ 2, 5/3, 4/3 ]; [ 3, 5/2 ]; 27*x/4 ).
G.f.: A(x) = G(x)^4 where G(x) = 1 + x*G(x)^3 = g.f. of A001764 giving a(n)=C(3n+m-1,n)*m/(2n+m) at power m=4 with offset n=0. - Paul D. Hanna, May 10 2008
G.f.: (((4*sin(arcsin((3*sqrt(3*x))/2)/3))/(sqrt(3*x))-1)^2-1)/(4*x). - Vladimir Kruchinin, Feb 17 2023
E.g.f.: hypergeom([4/3, 5/3, 2]; [1, 5/2, 3]; 27*x/4). - G. C. Greubel, Aug 29 2025
a(n) ~ 3^(3*n+7/2) / (2^(2*n+3) * n^(3/2) * sqrt(Pi)). - Amiram Eldar, Sep 12 2025
O.g.f.: (hypergeom([1/3, 2/3], [3/2], (27*x)/4))^4, denoted by h(x), satisfies -1 + (-4*x + 1)*h(x) - 2*x^2*h(x)^2 + x^4*h(x)^3 = 0.
O.g.f.: (16*(-(sqrt(4 - 27*x)/2 + 3*i*sqrt(3)*sqrt(x)/2)^(1/3)/2 + (sqrt(4 - 27*x)/2 - 3*i*sqrt(3)*sqrt(x)/2)^(1/3)/2)^4)/(9*x^2), where i = sqrt(-1), the imaginary unit. - Karol A. Penson, Sep 14 2025
MATHEMATICA
Table[2*Binomial[3*n+3, n]/(n+2), {n, 0, 40}] (* G. C. Greubel, Aug 29 2025 *)
PROG
(PARI) a(n)=my(m=4); binomial(3*n+m-1, n)*m/(2*n+m) /* 4th power of A001764 with offset n=0 */ \\ Paul D. Hanna, May 10 2008
(Magma)
A006629:= func< n | 2*Binomial(3*n+3, n)/(n+2) >;
[A006629(n): n in [0..40]]; // G. C. Greubel, Aug 29 2025
(SageMath)
def A006629(n): return 2*binomial(3*n+3, n)//(n+2)
print([A006629(n) for n in range(41)]) # G. C. Greubel, Aug 29 2025
CROSSREFS
KEYWORD
nonn,easy,changed
AUTHOR
EXTENSIONS
More precise definition from Paul D. Hanna, May 10 2008
STATUS
approved