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A026004 a(n) = T(3n+1,n), where T = Catalan triangle (A008315). 6
1, 3, 14, 75, 429, 2548, 15504, 95931, 600875, 3798795, 24192090, 154969620, 997490844, 6446369400, 41802112192, 271861216539, 1772528290407, 11582393855305, 75831424919250, 497337483739635, 3266814940064445 (list; graph; refs; listen; history; text; internal format)
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

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Last modified October 23 16:46 EDT 2019. Contains 328373 sequences. (Running on oeis4.)