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A174687 Central coefficients T(2n,n) of the Catalan triangle A033184. 9
1, 2, 9, 48, 275, 1638, 9996, 62016, 389367, 2466750, 15737865, 100975680, 650872404, 4211628008, 27341497800, 177996090624, 1161588834303, 7596549816030, 49772989810635, 326658445806000, 2147042307851595, 14130873926790390, 93115841412899760 (list; graph; refs; listen; history; text; internal format)
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
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
Column k=2 of A214776.
Sequence in context: A047139 A190315 A190253 * A047059 A153297 A364734
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Mar 27 2010
STATUS
approved

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Last modified February 29 01:59 EST 2024. Contains 370401 sequences. (Running on oeis4.)