|
%I #115 Dec 14 2023 14:54:16
%S 1,4,21,120,715,4368,27132,170544,1081575,6906900,44352165,286097760,
%T 1852482996,12033222880,78378960360,511738760544,3348108992991,
%U 21945588357420,144079707346575,947309492837400,6236646703759395
%N a(n) = binomial(3*n+1,n).
%C Number of leaves in all noncrossing rooted trees on n nodes on a circle.
%C Number of standard tableaux of shape (n-1,1^(2n-3)). - _Emeric Deutsch_, May 25 2004
%C 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
%C a(n+2) gives row sums of A119301. - _Paul Barry_, May 13 2006
%C 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
%C Central coefficients of triangle A078812. - _Vladimir Kruchinin_, May 10 2012
%C Row sums of A252501. - _L. Edson Jeffery_, Dec 18 2014
%H Vincenzo Librandi, <a href="/A045721/b045721.txt">Table of n, a(n) for n = 0..200</a>
%H Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative Functions</a>
%H D. Kruchinin and V. Kruchinin, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Kruchinin/kruchinin5.html">A Method for Obtaining Generating Function for Central Coefficients of Triangles</a>, Journal of Integer Sequence, Vol. 15 (2012), article 12.9.3.
%H W. Mlotkowski and K. A. Penson, <a href="http://arxiv.org/abs/1309.0595">Probability distributions with binomial moments</a>, arXiv preprint arXiv:1309.0595 [math.PR], 2013.
%H <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%F a(n) is asymptotic to c/sqrt(n)*(27/4)^n with c=0.73... - _Benoit Cloitre_, Jan 27 2003
%F 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
%F a(n+2) = C(3n+1,n) = Sum_{k=0..n} C(3n-k,n-k). - _Paul Barry_, May 13 2006
%F a(n+2) = C(3n+1,2n+1) = A078812(2n,n). - _Paul Barry_, Nov 09 2006
%F 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
%F From _Peter Luschny_, Sep 04 2012: (Start)
%F O.g.f.: hypergeometric2F1([2/3, 4/3], [3/2], x*27/4).
%F a(n) = (n+1)*hypergeometric2F1([-2*n, -n], [2], 1). (End)
%F 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
%F a(n) = Sum_{r=0..n} C(n,r) * C(2*n+1,r). - _J. M. Bergot_, Mar 18 2014
%F From _Peter Bala_, Nov 04 2015: (Start)
%F a(n) = Sum_{k = 0..n} 1/(2*k + 1)*binomial(3*n - 3*k,n - k)*binomial(3*k, k).
%F 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)
%F a(n) = [x^n] 1/(1 - x)^(2*(n+1)). - _Ilya Gutkovskiy_, Oct 10 2017
%F O.g.f.: (i/24)*((4*sqrt(4 - 27*z) + 12*i*sqrt(3)*sqrt(z))^(2/3) - (4*sqrt(4 - 27*z) - 12*i*sqrt(3)*sqrt(z))^(2/3))*sqrt(3)*8^(1/3)*sqrt(4 - 27*z)/(sqrt(z)*(-4 + 27*z)), where i = sqrt(-1). - _Karol A. Penson_, Dec 13 2023
%p [seq( binomial(3*n+1,n),n=0..40)]; # _N. J. A. Sloane_, Jun 09 2007
%t Table[Binomial[3 n + 1, n], {n, 0, 20}] (* _Vincenzo Librandi_, Aug 07 2014 *)
%o (PARI) a(n)=binomial(3*n+1,n) \\ _Charles R Greathouse IV_, Mar 18 2014
%o (Magma) [Binomial(3*n+1, n): n in [0..20]]; // _Vincenzo Librandi_, Aug 07 2014
%Y Cf. A252501, A263134 (partial sums), A001764, A004319, A005809, A006013, A013698, A025174, A117671, A165817, A236194.
%K nonn,easy
%O 0,2
%A _Emeric Deutsch_
%E 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.
| 