A006629 Self-convolution 4th power of A001764, which enumerates ternary trees. (Formerly M3542)

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

Offset

0,2

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

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.: 3_F_2 ( [ 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

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

Crossrefs

Column 2 of A092276.

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

Cf. A000139.

Sequence in context: A081671 A244785 A260650 * A068764 A127394 A046984

Adjacent sequences: A006626 A006627 A006628 * A006630 A006631 A006632

Keyword

nonn,easy

Author

Simon Plouffe, N. J. A. Sloane

Extensions

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

Status

approved