A005790 4-dimensional Catalan numbers. (Formerly M4954)

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

Offset

0,3

Comments

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

References

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.

Links

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).

K. Gorska and K. A. Penson, Multidimensional Catalan and related numbers as Hausdorff moments, arXiv preprint arXiv:1304.6008 [math.CO], 2013.

S. Snover, Letter to N. J. A. Sloane, May 1991

S. F. Troyer & S. L. Snover, m-Dimensional Catalan numbers, Preprint, 1989. (Annotated scanned copy)

Formula

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

Maple

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

Mathematica

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

Prog

(Magma) [12*Factorial(4*n)/(Factorial(n)*Factorial(n+1)*Factorial(n+2) *Factorial(n+3)): n in [0..20]]; // Vincenzo Librandi, Nov 23 2018
(PARI) vector(20, n, n--; 12*(4*n)!/(n!*(n+1)!*(n+2)!*(n+3)!)) \\ G. C. Greubel, Nov 23 2018
(Sage) [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

Crossrefs

A row of A060854.

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

Sequence in context: A319096 A297548 A215787 * A208563 A200061 A171208

Adjacent sequences: A005787 A005788 A005789 * A005791 A005792 A005793

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

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

Status

approved