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 A005789 3-dimensional Catalan numbers. (Formerly M3997) 26
 1, 1, 5, 42, 462, 6006, 87516, 1385670, 23371634, 414315330, 7646001090, 145862174640, 2861142656400, 57468093927120, 1178095925505960, 24584089974896430, 521086299271824330, 11198784501894470250, 243661974372798631650, 5360563436201569896300 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Number of standard tableaux of shape (n,n,n). - Emeric Deutsch, May 13 2004 Number of walks within N^2 (the first quadrant of Z^2) starting and ending at (0,0) and consisting of 3 n steps taken from {(-1, 0), (0, 1), (1, -1)}. -  Manuel Kauers, Nov 18 2008 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, ...}.) - 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 3-dimensions 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 four-dimensional Catalan formula. Proceedings of the Nineteenth Manitoba Conference on Numerical Mathematics and Computing (Winnipeg, MB, 1989). Congr. Numer. 75 (1990), 123-126. LINKS Alois P. Heinz, Table of n, a(n) for n = 0..700 Nicolas Borie, Three-dimensional Catalan numbers and product-coproduct prographs, arXiv:1704.00212 [math.CO], 2017. M. Bousquet-Mélou and M. Mishna, Walks with small steps in the quarter plane, arXiv:0810.4387 [math.CO], 2008-2009. Paul Drube, Maxwell Krueger, Ashley Skalsky, Meghan Wren, Set-Valued Young Tableaux and Product-Coproduct Prographs, arXiv:1710.02709 [math.CO], 2017. 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 [math.CO], 2013. Martin Griffiths and Nick Lord, The hook-length formula and generalised Catalan numbers, The Mathematical Gazette Vol. 95, No. 532 (March 2011), pp. 23-30 R. Kenyon, J. Miller, S. Sheffield, D. B. Wilson, Bipolar orientations on planar maps and SLE_12, arXiv preprint arXiv:1511.04068 [math.PR], 2015. Also The Annals of Probability (2019) Vol. 47, No. 3, 1240-1269. J. B. Lewis, Pattern avoidance and RSK-like algorithms for alternating permutations and Young tableaux, arXiv:0909.4966 [math.CO], 2009-2011. [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 Andrew Lohr, Several Topics in Experimental Mathematics, arXiv:1805.00076 [math.CO], 2018. S. Snover, Letter to N. J. A. Sloane, May 1991 R. A. Sulanke, Three-dimensional 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) S. F. Troyer & S. L. Snover, m-Dimensional Catalan numbers, Preprint, 1989. (Annotated scanned copy) Wolfgang Unger, Combinatorics of Lattice QCD at Strong Coupling, arXiv:1411.4493 [hep-lat], 2014. Manuel Wettstein, Trapezoidal Diagrams, Upward Triangulations, and Prime Catalan Numbers, arXiv:1602.07235 [cs.CG], 2016 and Discr. Comp. Geom. 58 (2017) 505-525. 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!*..*(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],,27*x)+(216*x+1)* hypergeom([4/3, 5/3],,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*n-1)*(3*n-2)*a(n-1)=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 E.g.f.: 2F2(1/3,2/3; 2,3; 27*x). - Ilya Gutkovskiy, Oct 13 2017 MAPLE a:= n-> (3*n)! *mul(i!/(n+i)!, i=0..2): seq(a(n), n=0..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 *) aux[i_Integer, j_Integer, n_Integer] := Which[Min[i, j, n] < 0 || Max[i, j] > n, 0, n == 0, KroneckerDelta[i, j, n], True, aux[i, j, n] = aux[-1 + i, 1 + j, -1 + n] + aux[i, -1 + j, -1 + n] + aux[1 + i, j, -1 + n]]; Table[aux[0, 0, 3 n], {n, 0, 25}] (* Manuel Kauers, Nov 18 2008 *) PROG (MAGMA) [2*Factorial(3*n)/(Factorial(n)*Factorial(n+1)*Factorial(n+2)): n in [0..20]]; // Vincenzo Librandi, Oct 14 2017 (PARI) a(n) = 2*(3*n)!/(n!*(n+1)!*(n+2)!); \\ Altug Alkan, Mar 14 2018 CROSSREFS A row of A060854. A subset of A025035. See A268538 for primitive terms. Sequence in context: A217805 A217808 A151334 * A217810 A217809 A317352 Adjacent sequences:  A005786 A005787 A005788 * A005790 A005791 A005792 KEYWORD nonn,easy,walk,nice AUTHOR EXTENSIONS Added a(0), merged A151334 into this one. - N. J. A. Sloane, Feb 24 2016 STATUS approved

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Last modified August 21 23:01 EDT 2019. Contains 326169 sequences. (Running on oeis4.)