A026004 a(n) = T(3n+1,n), where T = Catalan triangle (A008315).

1, 3, 14, 75, 429, 2548, 15504, 95931, 600875, 3798795, 24192090, 154969620, 997490844, 6446369400, 41802112192, 271861216539, 1772528290407, 11582393855305, 75831424919250, 497337483739635, 3266814940064445

Offset

0,2

Comments

Number of standard tableaux of shape (2n+1,n). Example: a(1)=3 because in the top row we can have 134, 124, or 123 (but not 234). - Emeric Deutsch, May 23 2004

Number of noncrossing forests with n+2 vertices and two components. - Emeric Deutsch, May 31 2004

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

P. Flajolet and M. Noy, Analytic combinatorics of noncrossing configurations, Discrete Math. 204 (1999), 203-229.

Formula

a(n) = (n+2)/(2*n+2) * C(3*n+1, n). - Ralf Stephan, Apr 30 2004

G.f.: ((sqrt(x)*sin(2/3*arcsin((3*sqrt(3)*sqrt(x))/2)))/sqrt(4/3-9*x)-cos(1/3*arccos(1-(27*x)/2))+1)/(3*x). - conjectured by Harvey P. Dale, Jun 30 2011

G.f.: (2*g-1)/((3*g-1)*(g-1)^2) where g*(1-g)^2 = x. - Mark van Hoeij, Nov 09 2011

2*(n+1)*(2*n+1)*a(n) +(-43*n^2-3*n+6)*a(n-1) +12*(3*n-2)*(3*n-4)*a(n-2)=0. - R. J. Mathar, Jun 07 2013

a(n) = sum(k=0..n, (k+1)*binomial(n,k)*binomial(2*(n+1),n-k))/(n+1). - Vladimir Kruchinin, Mar 01 2014

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

Mathematica

Table[(n+2)/(2n+2)Binomial[3n+1, n], {n, 0, 20}] (* Harvey P. Dale, Jun 29 2011 *)

Prog

(Maxima) a(n):=sum((k+1)*binomial(n, k)*binomial(2*(n+1), n-k), k, 0, n)/(n+1); /* Vladimir Kruchinin, Mar 01 2014 */
(PARI) a(n) = (n+2)/(2*n+2) * binomial(3*n+1, n); \\ Joerg Arndt, Mar 01 2014

Crossrefs

Cf. A045722.

Sequence in context: A245246 A126122 A303034 * A200718 A063016 A246455

Adjacent sequences: A026001 A026002 A026003 * A026005 A026006 A026007

Keyword

nonn

Author

Clark Kimberling

Extensions

More terms from Ralf Stephan, Apr 30 2004

Status

approved