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A005790 4-dimensional Catalan numbers.
(Formerly M4954)
4
1, 14, 462, 24024, 1662804, 140229804, 13672405890, 1489877926680, 177295473274920, 22661585038594320, 3073259571003214320, 438091463242348309440, 65166105157299311029200, 10056663345892631910888600, 1602608179958939072505281850, 262708662267696303439658400600 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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

Alois P. Heinz, Table of n, a(n) for n = 1..130

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.

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(4n)!/(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=1..20);  # Alois P. Heinz, Jul 25 2012

MATHEMATICA

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

CROSSREFS

A row of A060854.

Sequence in context: A282245 A297548 A215787 * A208563 A200061 A171208

Adjacent sequences:  A005787 A005788 A005789 * A005791 A005792 A005793

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified August 20 03:10 EDT 2018. Contains 313904 sequences. (Running on oeis4.)