1,2

Number of standard tableaux of shape (n,n,n). - Emeric Deutsch (deutsch(AT)duke.poly.edu), May 13 2004

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]

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 four-dimensional Catalan formula. Proceedings of the Nineteenth Manitoba Conference on Numerical Mathematics and Computing (Winnipeg, MB, 1989). Congr. Numer. 75 (1990), 123-126.

R. A. Sulanke, Generalizing Narayana and Schroeder Numbers to Higher Dimensions, Electron. J. Combin. 11 (2004), Research Paper 54, 20 pp.

R. A. Sulanke, Three-dimensional Narayana and Schr\"oder numbers

J. B. Lewis, Pattern avoidance and RSK-like algorithms for alternating permutations and Young tableaux [From Joel Brewster Lewis (jblewis(AT)post.harvard.edu), Oct 04 2009]

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.

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]

(* first do *) Needs["DiscreteMath`Combinatorica`"] (* then *) Table[ NumberOfTableaux@ {n, n, n}], {n, 17}] (* Robert G. Wilson v Nov 15 2006 *)

A row of A060854.

Sequence in context: A082145 A126765 A024492 this_sequence A151334 A102693 A052654

Adjacent sequences: A005786 A005787 A005788 this_sequence A005790 A005791 A005792

nonn,easy

N. J. A. Sloane (njas(AT)research.att.com).