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URL: https://oeis.org/A026004

⇱ A026004 - OEIS

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A026004
a(n) = T(3n+1,n), where T = Catalan triangle (A008315).
7
1, 3, 14, 75, 429, 2548, 15504, 95931, 600875, 3798795, 24192090, 154969620, 997490844, 6446369400, 41802112192, 271861216539, 1772528290407, 11582393855305, 75831424919250, 497337483739635, 3266814940064445, 21488271095284560, 141521997156845760, 933129303062092500
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
Philippe Flajolet and Marc Noy, Analytic combinatorics of noncrossing configurations, Discrete Math., Vol. 204, No. 1-3 (1999), 203-229.
FORMULA
a(n) = (n+2)/(2*n+2) * binomial(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
a(n) ~ 3^(3*n+3/2) / (2^(2*n+3) * sqrt(Pi*n)). - Amiram Eldar, Nov 06 2025
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 A391649
KEYWORD
nonn,easy
EXTENSIONS
More terms from Ralf Stephan, Apr 30 2004
STATUS
approved