A006629 Self-convolution 4th power of A001764, which enumerates ternary trees.

1, 4, 18, 88, 455, 2448, 13566, 76912, 444015, 2601300, 15426840, 92431584, 558685348, 3402497504, 20858916870, 128618832864, 797168807855, 4963511449260, 31032552351570, 194743066471800, 1226232861415695

Offset: 0

Author

Simon Plouffe, N. J. A. Sloane

Keywords

• nonn
• easy
• changed

Comments

Sum of root degrees of all noncrossing trees on nodes on a circle. - Emeric Deutsch

References

• 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.
• 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
• 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.
• C. H. Pah, Single polygon counting on Cayley Tree of order 3, J. Stat. Phys. 140 (2010) 198-207.
• Index entries for sequences related to rooted trees

Crossrefs

Column 2 of A092276.

Cf. A000139, A001764, A006013, A006630, A006631, A230547, A233657.

Programs

Magma

A006629:= func< n | 2*Binomial(3*n+3,n)/(n+2) >;
[A006629(n): n in [0..40]]; // G. C. Greubel, Aug 29 2025

Mathematica

Table[2*Binomial[3*n+3,n]/(n+2), {n,0,40}] (* G. C. Greubel, Aug 29 2025 *)

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

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

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

Extensions

More precise definition from Paul D. Hanna, May 10 2008