%I #51 Feb 06 2024 01:40:28

%S 1,8,55,364,2380,15504,100947,657800,4292145,28048800,183579396,

%T 1203322288,7898654920,51915526432,341643774795,2250829575120,

%U 14844575908435,97997533741800,647520696018735,4282083008118300

%N a(n) = binomial(3*n+2, n-1).

%C Degree of variety K_{2,n}^1. Also number of double-rises (or odd-level peaks) in all generalized {(1,2),(1,-1)}-Dyck paths of length 3(n+1).

%C Number of dissections of a convex (2n+2)-gon by n-2 noncrossing diagonals into (2j+2)-gons, 1<=j<=n-1.

%C a(n) is the number of lattice paths from (0,0) to (3n+1,n-1) avoiding two consecutive up-steps. - _Shanzhen Gao_, Apr 20 2010

%H Robert Israel, <a href="/A013698/b013698.txt">Table of n, a(n) for n = 1..1000</a>

%H M. S. Ravi et al., <a href="http://dx.doi.org/10.1137/S036301299325270X">Dynamic Pole Assignment and Schubert Calculus</a>, SIAM J. Control Optimization, 34 (1996), 813-832, esp. p. 825.

%H Daniel W. Stasiuk, <a href="http://hdl.handle.net/10388/11865">An Enumeration Problem for Sequences of n-ary Trees Arising from Algebraic Operads</a>, Master's Thesis, University of Saskatchewan-Saskatoon (2018).

%F G.f.: g/((g-1)^3*(3*g-1)) where g*(1-g)^2 = x. - _Mark van Hoeij_, Nov 09 2011

%F a(n) = Sum_{k=0..n-1} binomial(2*n+k+2,k). - _Arkadiusz Wesolowski_, Apr 02 2012

%F D-finite with recurrence 2*(2*n+3)*(n+1)*a(n) -n*(67*n+34)*a(n-1) +30*(3*n-1)*(3*n-2)*a(n-2)=0. - _R. J. Mathar_, Feb 05 2013

%F a(n+1) = (3*n+5)*(3*n+4)*(3*n+3)*a(n)/((2*n+5)*(2*n+4)*n). - _Robert Israel_, Aug 09 2015

%F With offset 0, the o.g.f. equals f(x)*g(x)^5, 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. A045721 (k = 1), A025174 (k = 2), A004319 (k = 3), A236194 (k = 4), A165817 (k = -1), A117671 (k = -2). - _Peter Bala_, Nov 04 2015

%p seq(binomial(3*n+2,n-1), n=0..30); # _Robert Israel_, Aug 09 2015

%t Table[Binomial[3*n+2, n-1], {n, 25}] (* _Arkadiusz Wesolowski_, Apr 02 2012 *)

%o (PARI) first(m)=vector(m,n,binomial(3*n+2, n-1)); /* _Anders Hellström_, Aug 09 2015 */

%o (Magma) [Binomial(3*n+2, n-1): n in [1..25]]; // _Vincenzo Librandi_, Aug 10 2015

%o (Sage) [binomial(3*n+2, n-1) for n in (1..25)] # _G. C. Greubel_, Mar 21 2019

%o (GAP) List([1..25], n-> Binomial(3*n+2, n-1)) # _G. C. Greubel_, Mar 21 2019

%Y Cf. A013699 (q=2), A013700 (q=3), A013701 (q=4), A013702 (q=5).

%Y A column of triangle A102537. Cf. A001764, A004319, A005809, A006013, A025174, A045721, A117671, A165817, A236194.

%K nonn,easy

%O 1,2

%A Joachim.Rosenthal(AT)nd.edu (Joachim Rosenthal), _Emeric Deutsch_