%I A005789 M3997
%S A005789 1,5,42,462,6006,87516,1385670,23371634,414315330,7646001090,
%T A005789 145862174640,2861142656400,57468093927120,1178095925505960,
%U A005789 24584089974896430,521086299271824330,11198784501894470250
%N A005789 3-dimensional Catalan numbers.
%C A005789 Number of standard tableaux of shape (n,n,n). - Emeric Deutsch (deutsch(AT)duke.poly.edu), 
               May 13 2004
%C A005789 Number of up-down permutations of length 2n with no four-term increasing 
               subsequence, or equivalently the number of down-up permutations of 
               length 2n with no four-term decreasing subsequence. (An up-down permutation 
               is one whose descent set is {2, 4, 6, ...}.) [From Joel Brewster 
               Lewis (jblewis(AT)post.harvard.edu), Oct 04 2009]
%D A005789 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A005789 Snover, Stephen L.; and Troyer, Stephanie F.; A four-dimensional Catalan 
               formula. Proceedings of the Nineteenth Manitoba Conference on Numerical 
               Mathematics and Computing (Winnipeg, MB, 1989). Congr. Numer. 75 
               (1990), 123-126.
%D A005789 R. A. Sulanke, Generalizing Narayana and Schroeder Numbers to Higher 
               Dimensions, Electron. J. Combin. 11 (2004), Research Paper 54, 20 
               pp.
%H A005789 R. A. Sulanke, <a href="http://math.boisestate.edu/~sulanke/recentpapindex.html">
               Three-dimensional Narayana and Schr\"oder numbers</a>
%H A005789 J. B. Lewis, <a href="http://arxiv.org/abs/0909.4966">Pattern avoidance 
               and RSK-like algorithms for alternating permutations and Young tableaux</
               a> [From Joel Brewster Lewis (jblewis(AT)post.harvard.edu), Oct 04 
               2009]
%F A005789 a(n) = 2*(3*n)!/(n!*(n+1)!*(n+2)!); a(n) = 0!*1!*..*(k-1)! *(k*n)! / 
               ( n!*(n+1)!*..*(n+k-1)! ) for k=3.
%F A005789 G.f. (1/30)*(1/x-27)*(9*hypergeom([1/3, 2/3],[1],27*x)+(216*x+1)*hypergeom([4/
               3, 5/3],[2],27*x))-1/(3*x) [From Mark van Hoeij (hoeij(AT)math.fsu.edu), 
               Oct 14 2009]
%t A005789 (* first do *) Needs["DiscreteMath`Combinatorica`"] (* then *) Table[ 
               NumberOfTableaux@ {n, n, n}], {n, 17}] (* Robert G. Wilson v Nov 
               15 2006 *)
%Y A005789 A row of A060854.
%Y A005789 Sequence in context: A082145 A126765 A024492 this_sequence A151334 A102693 
               A052654
%Y A005789 Adjacent sequences: A005786 A005787 A005788 this_sequence A005790 A005791 
               A005792
%K A005789 nonn,easy
%O A005789 1,2
%A A005789 N. J. A. Sloane (njas(AT)research.att.com).