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 A045721 a(n) = binomial(3*n+1,n). 27
 1, 4, 21, 120, 715, 4368, 27132, 170544, 1081575, 6906900, 44352165, 286097760, 1852482996, 12033222880, 78378960360, 511738760544, 3348108992991, 21945588357420, 144079707346575, 947309492837400, 6236646703759395 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Number of leaves in all noncrossing rooted trees on n nodes on a circle. Number of standard tableaux of shape (n-1,1^(2n-3)). - Emeric Deutsch, May 25 2004 a(n) = number of Dyck (2n-3)-paths with exactly one descent of odd length. For example, a(3) counts all 5 Dyck 3-paths except UDUDUD. - David Callan, Jul 25 2005 a(n+2) gives row sums of A119301. - Paul Barry, May 13 2006 a(n) is the number of paths avoiding UU from (0,0) to (3n,n) and taking steps from {U,H}. - Shanzhen Gao, Apr 15 2010 Central coefficients of triangle A078812. - Vladimir Kruchinin, May 10 2012 Row sums of A252501. - L. Edson Jeffery, Dec 18 2014 LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..200 Milan Janjic, Two Enumerative Functions 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. W. Mlotkowski and K. A. Penson, Probability distributions with binomial moments, arXiv preprint arXiv:1309.0595 [math.PR], 2013. FORMULA a(n) is asymptotic to c/sqrt(n)*(27/4)^n with c=0.73... - Benoit Cloitre, Jan 27 2003 G.f.: gz^2/(1-3zg^2), where g=g(z) is given by g=1+zg^3, g(0)=1, i.e. (in Maple command) g := 2*sin(arcsin(3*sqrt(3*z)/2)/3)/sqrt(3*z); - Emeric Deutsch, May 22 2003 a(n+2) = C(3n+1,n) = Sum_{k=0..n} C(3n-k,n-k). - Paul Barry, May 13 2006 a(n+2) = C(3n+1,2n+1) = A078812(2n,n); - Paul Barry, Nov 09 2006 G.f.: A(x)=(2*cos(asin((3^(3/2)*sqrt(x))/2)/3)* sin(asin((3^(3/2)* sqrt(x))/2)/3))/(sqrt(3)*sqrt(1-(27*x)/4)*sqrt(x)). - Vladimir Kruchinin, Jun 10 2012 O.g.f.: hypergeometric2F1([2/3, 4/3], [3/2], x*27/4). a(n) = (n+1)*hypergeometric2F1([-2*n, -n], , 1). - Peter Luschny, Sep 04 2012 D-finite with recurrence 2*n*(2*n+1)*a(n) - 3*(3*n-1)*(3*n+1)*a(n-1) = 0. - R. J. Mathar, Feb 05 2013 a(n) = Sum_{r=0..n} C(n,r) * C(2*n+1,r). - J. M. Bergot, Mar 18 2014 From Peter Bala, Nov 04 2015: (Start) a(n) = binomial(3*n + 1,n). a(n) = Sum_{k = 0..n} 1/(2*k + 1)*binomial(3*n - 3*k,n - k)*binomial(3*k, k). O.g.f. equals f(x)*g(x), where f(x) is the o.g.f. for A005809 and g(x) is the o.g.f. for A001764. More generally, f(x)*g(x)^k is the o.g.f. for the sequence binomial(3*n + k,n). Cf. A025174 (k = 2), A004319 (k = 3), A236194 (k = 4), A013698 (k = 5), A165817 (k = -1), A117671 (k = -2). (End) a(n) = [x^n] 1/(1 - x)^(2*(n+1)). - Ilya Gutkovskiy, Oct 10 2017 MAPLE [seq( binomial(3*n+1, n), n=0..40)]; # N. J. A. Sloane, Jun 09 2007 MATHEMATICA Table[Binomial[3 n + 1, n], {n, 0, 20}] (* Vincenzo Librandi, Aug 07 2014 *) PROG (PARI) a(n)=binomial(3*n+1, n) \\ Charles R Greathouse IV, Mar 18 2014 (MAGMA) [Binomial(3*n+1, n): n in [0..20]]; // Vincenzo Librandi, Aug 07 2014 CROSSREFS Cf. A252501, A263134 (partial sums), A001764, A004319, A005809, A006013, A013698, A025174, A117671, A165817, A236194. Sequence in context: A093426 A046090 A182435 * A101810 A274969 A236525 Adjacent sequences:  A045718 A045719 A045720 * A045722 A045723 A045724 KEYWORD nonn,easy AUTHOR EXTENSIONS Simpler definition from Ira M. Gessel, May 26 2007. This change means that most of the offsets in the comments will now need to be changed too. STATUS approved

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Last modified September 26 16:43 EDT 2020. Contains 337374 sequences. (Running on oeis4.)