A268538 -id:A268538 - cp's OEIS frontend

A005789 3-dimensional Catalan numbers.

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

Offset: 0

N. J. A. Sloane

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 [here "count" is meant as "number of symbols in any prefix", Joerg Arndt, Jan 02 2024]

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.

Alois P. Heinz, Table of n, a(n) for n = 0..700
Joerg Arndt, The a(3)=42 Young tableaux of shape [3,3,3].
Nicolas Borie, Three-dimensional Catalan numbers and product-coproduct prographs, arXiv:1704.00212 [math.CO], 2017.
Mireille Bousquet-Mélou and Marni Mishna, Walks with small steps in the quarter plane, arXiv:0810.4387 [math.CO], 2008-2009.
Paul Drube, Maxwell Krueger, Ashley Skalsky, and 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
Katarzyna Górska and Karol 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
Richard Kenyon, Jason Miller, Scott Sheffield, and David 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.
Joel Brewster 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]
Joel Brewster 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.
Michaël Moortgat, The Tamari order for D^3 and derivability in semi-associative Lambek-Grishin Calculus, 15th Workshop: Computational Logic and Applications (CLA 2020).
Dimana Miroslavova Pramatarova, Investigating the Periodicity of Weighted Catalan Numbers and Generalizing Them to Higher Dimensions, MIT Res. Sci. Instit. (2025). See p. 13.
Maya Sankar, Further Bijections to Pattern-Avoiding Valid Hook Configurations, arXiv:1910.08895 [math.CO], 2019.
Stephen Snover, Letter to N. J. A. Sloane, May 1991
Robert A. Sulanke, Three-dimensional Narayana and Schröder numbers, Theoretical Computer Science, Volume 346, Issues 2-3, 28 November 2005, Pages 455-468.
Robert A. Sulanke, Generalizing Narayana and Schroeder Numbers to Higher Dimensions, Electron. J. Combin. 11 (2004), Research Paper 54, 20 pp. (see page 16)
Stephanie F. Troyer and Stephen 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.

A row of A060854.
A subset of A025035.
See A268538 for primitive terms.

[2*Factorial(3*n)/(Factorial(n)*Factorial(n+1)*Factorial(n+2)): n in [0..20]]; // Vincenzo Librandi, Oct 14 2017

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

Needs["Combinatorica`"]
Table[ NumberOfTableaux@ {n, n, n}, {n, 0, 17}] (* Robert G. Wilson v, Nov 15 2006 *)
Table[2*(3*n)!/(n!*(n+1)!*(n+2)!),{n,0,20}] (* 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 *)

a(n) = 2*(3*n)!/(n!*(n+1)!*(n+2)!); \\ Altug Alkan, Mar 14 2018

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). - 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
D-finite with recurrence (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

Added a(0), merged A151334 into this one. - N. J. A. Sloane, Feb 24 2016

A005790 4-dimensional Catalan numbers.

Original entry on oeis.org

1, 1, 14, 462, 24024, 1662804, 140229804, 13672405890, 1489877926680, 177295473274920, 22661585038594320, 3073259571003214320, 438091463242348309440, 65166105157299311029200, 10056663345892631910888600, 1602608179958939072505281850, 262708662267696303439658400600

Offset: 0

N. J. A. Sloane

Number of standard tableaux of shape (n,n,n,n). - Emeric Deutsch, May 13 2004

The prime terms (as defined in A268538) are 1, 1, 10, 320, 16764, 1171355, 99315236, 9691755128, 1053114415100, ... - R. J. Mathar, Feb 27 2018

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Snover, Stephen L.; 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.

Seiichi Manyama, Table of n, a(n) for n = 0..423 (terms 1..130 from Alois P. Heinz)
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
Michaël Moortgat, The Tamari order for D^3 and derivability in semi-associative Lambek-Grishin Calculus, 15th Workshop: Computational Logic and Applications (CLA 2020).
Katarzyna Górska and Karol A. Penson, Multidimensional Catalan and related numbers as Hausdorff moments, arXiv preprint arXiv:1304.6008 [math.CO], 2013.
Dimana Miroslavova Pramatarova, Investigating the Periodicity of Weighted Catalan Numbers and Generalizing Them to Higher Dimensions, MIT Res. Sci. Instit. (2025). See p. 13.
Stephen Snover, Letter to N. J. A. Sloane, May 1991
Stephanie F. Troyer and Stephen L. Snover, m-Dimensional Catalan numbers, Preprint, 1989. (Annotated scanned copy)

A row of A060854.

Cf. A000108 (Catalan numbers), A005789, A005791.

[12*Factorial(4*n)/(Factorial(n)*Factorial(n+1)*Factorial(n+2) *Factorial(n+3)): n in [0..20]]; // Vincenzo Librandi, Nov 23 2018

a:= n-> (4*n)! * mul(i!/(4+i)!, i=0..n-1):
seq(a(n), n=0..20);  # Alois P. Heinz, Jul 25 2012

Table[12*(4*n)!/(n!*(n+1)!*(n+2)!*(n+3)!), {n, 0, 20}] (* Vaclav Kotesovec, Nov 18 2016 *)

vector(20, n, n--; 12*(4*n)!/(n!*(n+1)!*(n+2)!*(n+3)!)) \\ G. C. Greubel, Nov 23 2018

[12*factorial(4*n)/(factorial(n)*factorial(n+1)*factorial(n+2) *factorial(n+3)) for n in range(20)] # G. C. Greubel, Nov 23 2018

a(n) = 12*(4*n)!/(n! *(n+1)! *(n+2)! *(n+3)!).
G.f.: 4_F_3 ( [ 1, 3/2, 5/4, 7/4 ]; [ 3, 4, 5 ]; 256 x ).
a(n) ~ 3*2^(8*n+3/2)/(Pi^(3/2)*n^(15/2)). - Vaclav Kotesovec, Nov 18 2016
E.g.f.: 3F3(1/4,1/2,3/4; 2,3,4; 256*x) - 1. - Ilya Gutkovskiy, Oct 13 2017
(n+3)*(n+2)*(n+1)*a(n) -8*(4*n-3)*(2*n-1)*(4*n-1)*a(n-1)=0. - R. J. Mathar, Mar 04 2018

a(0)=1 prepended by Seiichi Manyama, Nov 23 2018

cp's OEIS Frontend

A005789 3-dimensional Catalan numbers.

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