A174687 Central coefficients T(2n,n) of the Catalan triangle A033184.

1, 2, 9, 48, 275, 1638, 9996, 62016, 389367, 2466750, 15737865, 100975680, 650872404, 4211628008, 27341497800, 177996090624, 1161588834303, 7596549816030, 49772989810635, 326658445806000, 2147042307851595, 14130873926790390, 93115841412899760

Offset: 0

Author

Paul Barry, Mar 27 2010

Keywords

• easy
• nonn
• changed

Comments

A033184 is the Riordan array (c(x), x*c(x)), c(x) the g.f. of A000108.

Number of standard Young tableaux of shape [2n, n]. Also the number of binary words with 2n 1's and n 0's such that for every prefix the number of 1's is >= the number of 0's. The a(2) = 9 words are: 101011, 101101, 101110, 110011, 110101, 110110, 111001, 111010, 111100. - Alois P. Heinz, Aug 15 2012

Number of lattice paths from (0,0) to (2n,n) not above y=x. - Ran Pan, Apr 08 2015

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..370
• Paul Barry, On the Central Coefficients of Riordan Matrices, Journal of Integer Sequences, 16 (2013), Article 13.5.1.
• Dmitry Kruchinin and Vladimir Kruchinin, A Method for Obtaining Generating Function for Central Coefficients of Triangles, Journal of Integer Sequence, Vol. 15 (2012), Article 12.9.3.
• Ran Pan, Exercise L, Project P
• Anssi Yli-Jyrä and Carlos Gómez-Rodríguez, Generic Axiomatization of Families of Noncrossing Graphs in Dependency Parsing, arXiv:1706.03357 [cs.CL], 2017.

Crossrefs

Cf. A000108, A001764, A033184.

Column k=2 of A214776.

Programs

Magma

[(n+1)*Binomial(3*n,n)/(2*n+1): n in [0..25]]; // Vincenzo Librandi, Apr 08 2015

Maple

a:= n-> binomial(3*n,n)*(n+1)/(2*n+1):
seq(a(n), n=0..25);  # Alois P. Heinz, Aug 15 2012

Mathematica

Table[Binomial[3n,n](n+1)/(2n+1), {n, 0, 25}] (* Vincenzo Librandi, Apr 08 2015 *)

PARI

a(n) = (n+1)*binomial(3*n,n)/(2*n+1); \\ Michel Marcus, Nov 12 2022

SageMath

[(n+1)*binomial(3*n,n)/(2*n+1) for n in range(31)] # G. C. Greubel, Nov 09 2022

Formula

a(n) = (n+1)*C(3*n, n)/(2n+1) = (n+1)*[x^(n+1)]( Rev(x/c(x)) ) = (n+1)*A001764(n), c(x) the g.f. of A000108.

G.f.: A(x) = sin(arcsin((3^(3/2)*sqrt(x))/2)/3)/(sqrt(3)*sqrt(x)) + cos(arcsin((3^(3/2)* sqrt(x))/2)/3)/(2*sqrt(1-(27*x)/4)). - Vladimir Kruchinin, May 25 2012

2*n*(2*n+1)*a(n) = 3*(13*n^2 -10*n +1)*a(n-1) -9*(3*n-4)*(3*n-5)*a(n-2). - R. J. Mathar, Nov 24 2012

a(n) = [x^n] ((1 - sqrt(1 - 4*x))/(2*x))^(n+1). - Ilya Gutkovskiy, Nov 01 2017

a(n) ~ 3^(3*n+1/2) / (4^(n+1) * sqrt(Pi*n)). - Amiram Eldar, Aug 29 2025