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

0,2

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), #13.5.1.

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.

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.

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

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 *)

Prog

(Magma) [(n+1)*Binomial(3*n, n)/(2*n+1): n in [0..25]]; // Vincenzo Librandi, Apr 08 2015
(SageMath) [(n+1)*binomial(3*n, n)/(2*n+1) for n in range(31)] # G. C. Greubel, Nov 09 2022
(PARI) a(n) = (n+1)*binomial(3*n, n)/(2*n+1); \\ Michel Marcus, Nov 12 2022

Crossrefs

Cf. A000108, A001764, A033184.

Column k=2 of A214776.

Sequence in context: A047139 A190315 A190253 * A047059 A153297 A364734

Adjacent sequences: A174684 A174685 A174686 * A174688 A174689 A174690

Keyword

easy,nonn

Author

Paul Barry, Mar 27 2010

Status

approved