

A005789


3dimensional Catalan numbers.
(Formerly M3997)


16



1, 5, 42, 462, 6006, 87516, 1385670, 23371634, 414315330, 7646001090, 145862174640, 2861142656400, 57468093927120, 1178095925505960, 24584089974896430, 521086299271824330, 11198784501894470250, 243661974372798631650, 5360563436201569896300
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OFFSET

1,2


COMMENTS

Number of standard tableaux of shape (n,n,n).  Emeric Deutsch, May 13 2004
Number of updown permutations of length 2n with no fourterm increasing subsequence, or equivalently the number of downup permutations of length 2n with no fourterm decreasing subsequence. (An updown permutation is one whose descent set is {2, 4, 6, ...}.)  Joel B. Lewis, Oct 04 2009
Equivalent to the number of standard tableaux: number of rectangular arrangements of [1..3n] into n increasing sequences of size 3 and 3 increasing sequences of size n. a(n) counts a subset of A025035(n).  Olivier Gérard, Feb 15 2011
Number of walks in 3dimensions using steps (1,0,0), (0,1,0), and (0,0,1) from (0,0,0) to (n,n,n) such that after each step we have z>=y>=x.  Thotsaporn Thanatipanonda, Feb 21 2012
Number of words consisting of n 'x' letters, n 'y' letters and n 'z' letters such that the 'x' count is always greater than or equal to the 'y' count and the 'y' count is always greater than or equal to the 'z' count; e.g., for n=2 we have xxyyzz, xxyzyz, xyxyzz, xyxzyz and xyzxyz.  Jon Perry, Nov 16 2012


REFERENCES

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Snover, Stephen L.; and Troyer, Stephanie F.; A fourdimensional Catalan formula. Proceedings of the Nineteenth Manitoba Conference on Numerical Mathematics and Computing (Winnipeg, MB, 1989). Congr. Numer. 75 (1990), 123126.


LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..700
Shalosh B. Ekhad and Doron Zeilberger, Computational and Theoretical Challenges on Counting Solid Standard Young Tableaux. Also arXiv preprint arXiv:1202.6229, 2012.  N. J. A. Sloane, Jul 07 2012
K. Gorska and K. A. Penson, Multidimensional Catalan and related numbers as Hausdorff moments, arXiv preprint arXiv:1304.6008, 2013
J. B. Lewis, Pattern avoidance and RSKlike algorithms for alternating permutations and Young tableaux [From Joel B. Lewis, Oct 04 2009]
J. B. Lewis, Pattern Avoidance for Alternating Permutations and Reading Words of Tableaux, Ph. D. Dissertation, Department of Mathematics, MIT, 2012.  From N. J. A. Sloane, Oct 12 2012
R. A. Sulanke, Threedimensional Narayana and Schröder numbers
R. A. Sulanke, Generalizing Narayana and Schroeder Numbers to Higher Dimensions, Electron. J. Combin. 11 (2004), Research Paper 54, 20 pp. (see page 16)
Wolfgang Unger, Combinatorics of Lattice QCD at Strong Coupling, arXiv:1411.4493 [heplat], 2014
Sherry H. F. Yan, On Wilf equivalence for alternating permutations, Elect. J. Combinat.; 20 (2013), #P58.


FORMULA

a(n) = 2*(3*n)!/(n!*(n+1)!*(n+2)!);
a(n) = 0!*1!*..*(k1)! *(k*n)! / ( n!*(n+1)!*..*(n+k1)! ) for k=3.
G.f. (1/30)*(1/x27)*(9*hypergeom([1/3, 2/3],[1],27*x)+(216*x+1)* hypergeom([4/3, 5/3],[2],27*x))1/(3*x).  Mark van Hoeij, Oct 14 2009
a(n) ~ 3^(3*n+1/2) / (Pi*n^4).  Vaclav Kotesovec, Nov 13 2014
a(n) = 2*A001700(n+1)*A001764(n+1)/(3*(3*n+1)*(3*n+2)).  R. J. Mathar, Aug 10 2015
(n+2)*(n+1)*a(n) 3*(3*n1)*(3*n2)*a(n1)=0.  R. J. Mathar, Aug 10 2015
G.f.: x*3F2(4/3,5/3,1;4,3;27x).  R. J. Mathar, Aug 10 2015


MAPLE

a:= n> (3*n)! *mul(i!/(n+i)!, i=0..2):
seq(a(n), n=1..20); # Alois P. Heinz, Feb 23 2012


MATHEMATICA

(* first do *) Needs["Combinatorica`"] (* then *) Table[ NumberOfTableaux@ {n, n, n}], {n, 17}] (* Robert G. Wilson v, Nov 15 2006 *)
Table[2*(3*n)!/(n!*(n+1)!*(n+2)!), {n, 1, 20}] (* Vaclav Kotesovec, Nov 13 2014 *)


CROSSREFS

A row of A060854.
A subset of A025035.
A151334 is apparently an essentially identical sequence.
Sequence in context: A024492 A217805 A217808 * A151334 A217809 A217810
Adjacent sequences: A005786 A005787 A005788 * A005790 A005791 A005792


KEYWORD

nonn,easy


AUTHOR

N. J. A. Sloane


STATUS

approved



