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%I #36 Nov 01 2017 17:11:09
%S 1,3,14,75,429,2548,15504,95931,600875,3798795,24192090,154969620,
%T 997490844,6446369400,41802112192,271861216539,1772528290407,
%U 11582393855305,75831424919250,497337483739635,3266814940064445
%N a(n) = T(3n+1,n), where T = Catalan triangle (A008315).
%C 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
%C Number of noncrossing forests with n+2 vertices and two components. - _Emeric Deutsch_, May 31 2004
%H Vincenzo Librandi, <a href="/A026004/b026004.txt">Table of n, a(n) for n = 0..200</a>
%H P. Flajolet and M. Noy, <a href="http://dx.doi.org/10.1016/S0012-365X(98)00372-0">Analytic combinatorics of noncrossing configurations</a>, Discrete Math. 204 (1999), 203-229.
%F a(n) = (n+2)/(2*n+2) * C(3*n+1, n). - _Ralf Stephan_, Apr 30 2004
%F 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
%F G.f.: (2*g-1)/((3*g-1)*(g-1)^2) where g*(1-g)^2 = x. - _Mark van Hoeij_, Nov 09 2011
%F 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
%F a(n) = sum(k=0..n, (k+1)*binomial(n,k)*binomial(2*(n+1),n-k))/(n+1). - _Vladimir Kruchinin_, Mar 01 2014
%F a(n) = [x^n] ((1 - sqrt(1 - 4*x))/(2*x))^(n+2). - _Ilya Gutkovskiy_, Nov 01 2017
%t Table[(n+2)/(2n+2)Binomial[3n+1,n],{n,0,20}] (* _Harvey P. Dale_, Jun 29 2011 *)
%o (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 */
%o (PARI) a(n) = (n+2)/(2*n+2) * binomial(3*n+1, n); \\ _Joerg Arndt_, Mar 01 2014
%Y Cf. A045722.
%K nonn
%O 0,2
%A _Clark Kimberling_
%E More terms from _Ralf Stephan_, Apr 30 2004
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