A233657 -id:A233657 - cp's OEIS frontend

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

Simon Plouffe, N. J. A. Sloane

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

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).

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

Column 2 of A092276.

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

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

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

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

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

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

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

A092276 Triangle read by rows: T(n,k) is the number of noncrossing trees with root degree equal to k.

Original entry on oeis.org

1, 2, 1, 7, 4, 1, 30, 18, 6, 1, 143, 88, 33, 8, 1, 728, 455, 182, 52, 10, 1, 3876, 2448, 1020, 320, 75, 12, 1, 21318, 13566, 5814, 1938, 510, 102, 14, 1, 120175, 76912, 33649, 11704, 3325, 760, 133, 16, 1, 690690, 444015, 197340, 70840, 21252, 5313, 1078, 168, 18, 1

Offset: 1

Emeric Deutsch, Feb 24 2004

With offset 0, Riordan array (f(x), x*f(x)) where f(x) is the g.f. of A006013. - Philippe Deléham, Jan 23 2010

			Triangle begins:
     1;
     2,    1;
     7,    4,    1;
    30,   18,    6,   1;
   143,   88,   33,   8,  1;
   728,  455,  182,  52, 10,  1;
  3876, 2448, 1020, 320, 75, 12, 1;
  ...
Top row of M^3 = (30, 18, 6, 1)
From _Peter Bala_, Nov 25 2024: (Start)
The transposed array as an infinite product of upper triangular arrays:
  /1 2 3 4 5 ... \/1            \/1              \       /1 2 7 30 143 ...\
  |  1 2 3 4 ... ||  1 2 3 4 ...||  1            |       |  1 4 18  88 ...|
  |    1 2 3 ... ||    1 2 3 ...||    1 2 3 4 ...| ... = |    1  6  33 ...|
  |      1 2 ... ||      1 2 ...||      1 2 3 ...|       |       1   8 ...|
  |        1 ... ||        1 ...||        1 2 ...|       |           1 ...|
  |          ... ||          ...||            ...|       |             ...|
Cf. A078812. (End)

Andrew Howroyd, Table of n, a(n) for n = 1..1275
Peter Bala, Factorisations of some Riordan arrays as infinite products
Paul Barry, Characterizations of the Borel triangle and Borel polynomials, arXiv:2001.08799 [math.CO], 2020.
Paul Barry, The second production matrix of a Riordan array, arXiv:2011.13985 [math.CO], 2020.
P. Flajolet and M. Noy, Analytic combinatorics of non-crossing configurations, Discrete Math., 204, 203-229, 1999.
M. Noy, Enumeration of noncrossing trees on a circle, Discrete Math., 180, 301-313, 1998.

Row sums give sequence A001764.

Columns 1..5 are A006013, A006629, A006630, A006631, A233657.

T := proc(n,k) if k=n then 1 else 2*k*binomial(3*n-k,n-k)/(3*n-k) fi end: seq(seq(T(n,k),k=1..n),n=1..11);

t[n_, n_] = 1; t[n_, k_] := 2*k*Binomial[3*n-k, n-k]/(3*n-k); Table[t[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Nov 22 2012, after Maple *)

T(n, k) = 2*k*binomial(3*n-k, n-k)/(3*n-k); \\ Andrew Howroyd, Nov 06 2017

T(n, k) = 2*k*binomial(3n-k, n-k)/(3n-k).
G.f.: 1/(1-t*z*g^2), where g := 2*sin(arcsin(3*sqrt(3*z)/2)/3)/sqrt(3*z) is the g.f. of the sequence A001764.
T(n, k) = Sum_{j>=1} j*T(n-1, k-2+j). - Philippe Deléham, Sep 14 2005
With offset 0, T(n,k) = ((n+1)/(k+1))*binomial(3n-k+1, n-k). - Philippe Deléham, Jan 23 2010
From Gary W. Adamson, Jul 07 2011: (Start)
Let M = the production matrix
  2, 1;
  3, 2, 1;
  4, 3, 2, 1;
  5, 4, 3, 2, 1;
  ...
Top row of M^(n-1) generates n-th row terms of triangle A092276. Leftmost terms of each row = A006013 starting (1, 2, 7, 30, 143, ...). (End)
Working with an offset of 0, the inverse array is the Riordan array ((1 - x)^2, x*(1 - x)^2). - Peter Bala, Apr 30 2024

A230547 a(n) = 3binomial(3n+9, n)/(n+3).

Original entry on oeis.org

1, 9, 63, 408, 2565, 15939, 98670, 610740, 3786588, 23535820, 146710476, 917263152, 5752004349, 36174046743, 228124619100, 1442387942520, 9142452842985, 58083251802345, 369816259792035, 2359448984037600

Offset: 0

Tim Fulford, Oct 23 2013

nonn

Fuss-Catalan sequence is a(n,p,r) = r*binomial(np+r,n)/(np+r), this is the case p=3, r=9.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
J-C. Aval, Multivariate Fuss-Catalan Numbers, arXiv:0711.0906 [math.CO], 2007; Discrete Math., 308 (2008), 4660-4669.
Thomas A. Dowling, Catalan Numbers Chapter 7
Wojciech Mlotkowski, Fuss-Catalan Numbers in Noncommutative Probability, Docum. Mathm. 15: 939-955.
Emanuele Munarini, Shifting Property for Riordan, Sheffer and Connection Constants Matrices, Journal of Integer Sequences, Vol. 20 (2017), Article 17.8.2.

Cf. A000108, A001764, A006013, A006629, A102893, A006630, A102594, A006631, A233657.

[9*Binomial(3*n+9, n)/(3*n+9): n in [0..30]];

Table[9 Binomial[3 n + 9, n]/(3 n + 9), {n, 0, 30}]

PARI
```
a(n) = 9*binomial(3*n+9,n)/(3*n+9);
```

{a(n)=local(B=1); for(i=0, n, B=(1+x*B^(3/9))^9+x*O(x^n)); polcoeff(B, n)}

G.f. satisfies: B(x) = {1 + x*B(x)^(p/r)}^r, here p=3, r=9.

D-finite with recurrence 2*n*(2*n+9)*(n+4)*a(n) -3*(3*n+7)*(n+2)*(3*n+8)*a(n-1)=0. - R. J. Mathar, Nov 22 2024

cp's OEIS Frontend

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

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A092276 Triangle read by rows: T(n,k) is the number of noncrossing trees with root degree equal to k.

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Maple

Mathematica

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Formula

A230547 a(n) = 3*binomial(3*n+9, n)/(n+3).

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Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

Formula

A230547 a(n) = 3binomial(3n+9, n)/(n+3).