A045721 a(n) = binomial(3*n+1,n).

1, 4, 21, 120, 715, 4368, 27132, 170544, 1081575, 6906900, 44352165, 286097760, 1852482996, 12033222880, 78378960360, 511738760544, 3348108992991, 21945588357420, 144079707346575, 947309492837400, 6236646703759395

Offset: 0

Author

Emeric Deutsch

Keywords

• nonn
• easy

Comments

Number of leaves in all noncrossing rooted trees on n nodes on a circle.

Number of standard tableaux of shape (n-1,1^(2n-3)). - Emeric Deutsch, May 25 2004

a(n) = number of Dyck (2n-3)-paths with exactly one descent of odd length. For example, a(3) counts all 5 Dyck 3-paths except UDUDUD. - David Callan, Jul 25 2005

a(n+2) gives row sums of A119301. - Paul Barry, May 13 2006

a(n) is the number of paths avoiding UU from (0,0) to (3n,n) and taking steps from {U,H}. - Shanzhen Gao, Apr 15 2010

Central coefficients of triangle A078812. - Vladimir Kruchinin, May 10 2012

Row sums of A252501. - L. Edson Jeffery, Dec 18 2014

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Milan Janjic, Two Enumerative Functions
• D. Kruchinin and V. Kruchinin, A Method for Obtaining Generating Function for Central Coefficients of Triangles, Journal of Integer Sequence, Vol. 15 (2012), article 12.9.3.
• W. Mlotkowski and K. A. Penson, Probability distributions with binomial moments, arXiv preprint arXiv:1309.0595 [math.PR], 2013.
• Index entries for sequences related to rooted trees

Crossrefs

Cf. A252501, A263134 (partial sums), A001764, A004319, A005809, A006013, A013698, A025174, A117671, A165817, A236194.

Programs

Magma

[Binomial(3*n+1, n): n in [0..20]]; // Vincenzo Librandi, Aug 07 2014

Maple

[seq( binomial(3*n+1,n),n=0..40)]; # N. J. A. Sloane, Jun 09 2007

Mathematica

Table[Binomial[3 n + 1, n], {n, 0, 20}] (* Vincenzo Librandi, Aug 07 2014 *)

PARI

a(n)=binomial(3*n+1,n) \\ Charles R Greathouse IV, Mar 18 2014

Formula

a(n) is asymptotic to c/sqrt(n)*(27/4)^n with c=0.73... - Benoit Cloitre, Jan 27 2003

G.f.: gz^2/(1-3zg^2), where g=g(z) is given by g=1+zg^3, g(0)=1, i.e. (in Maple command) g := 2*sin(arcsin(3*sqrt(3*z)/2)/3)/sqrt(3*z). - Emeric Deutsch, May 22 2003

a(n+2) = C(3n+1,n) = Sum_{k=0..n} C(3n-k,n-k). - Paul Barry, May 13 2006

a(n+2) = C(3n+1,2n+1) = A078812(2n,n). - Paul Barry, Nov 09 2006

G.f.: A(x)=(2*cos(asin((3^(3/2)*sqrt(x))/2)/3)* sin(asin((3^(3/2)* sqrt(x))/2)/3))/(sqrt(3)*sqrt(1-(27*x)/4)*sqrt(x)). - Vladimir Kruchinin, Jun 10 2012

From Peter Luschny, Sep 04 2012: (Start)

O.g.f.: hypergeometric2F1([2/3, 4/3], [3/2], x*27/4).

a(n) = (n+1)*hypergeometric2F1([-2*n, -n], [2], 1). (End)

D-finite with recurrence 2*n*(2*n+1)*a(n) - 3*(3*n-1)*(3*n+1)*a(n-1) = 0. - R. J. Mathar, Feb 05 2013

a(n) = Sum_{r=0..n} C(n,r) * C(2*n+1,r). - J. M. Bergot, Mar 18 2014

From Peter Bala, Nov 04 2015: (Start)

a(n) = Sum_{k = 0..n} 1/(2*k + 1)*binomial(3*n - 3*k,n - k)*binomial(3*k, k).

O.g.f. equals f(x)*g(x), where f(x) is the o.g.f. for A005809 and g(x) is the o.g.f. for A001764. More generally, f(x)*g(x)^k is the o.g.f. for the sequence binomial(3*n + k,n). Cf. A025174 (k = 2), A004319 (k = 3), A236194 (k = 4), A013698 (k = 5), A165817 (k = -1), A117671 (k = -2). (End)

a(n) = [x^n] 1/(1 - x)^(2*(n+1)). - Ilya Gutkovskiy, Oct 10 2017

O.g.f.: (i/24)*((4*sqrt(4 - 27*z) + 12*i*sqrt(3)*sqrt(z))^(2/3) - (4*sqrt(4 - 27*z) - 12*i*sqrt(3)*sqrt(z))^(2/3))*sqrt(3)*8^(1/3)*sqrt(4 - 27*z)/(sqrt(z)*(-4 + 27*z)), where i = sqrt(-1). - Karol A. Penson, Dec 13 2023

a(n-1) = (1/(4*n))*binomial(2*n, n)^2 * (1 - 3*((n - 1)/(n + 1))^3 + 5*((n - 1)*(n - 2)/((n + 1)*(n + 2)))^3 - 7*((n - 1)*(n - 2)*(n - 3)/((n + 1)*(n + 2)*(n + 3)))^3 + - ...) for n >= 1. Cf. A112029. - Peter Bala, Aug 08 2024

Extensions

Simpler definition from Ira M. Gessel, May 26 2007. This change means that most of the offsets in the comments will now need to be changed too.