A005791 5-dimensional Catalan numbers. (Formerly M5281)

1, 1, 42, 6006, 1662804, 701149020, 396499770810, 278607172289160, 231471904322784840, 219738059326729823880, 232553551737813227594400, 269396678720275351794712800, 336839101096824285057473785200, 449620757769949216266129125515200

Offset

0,3

Comments

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

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 = 0..294

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, and Prob. Math. Stat. 33 (2) (2013) 265-274.

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) = 0!*1!*..*(k-1)! *(k*n)! / ( n!*(n+1)!*..*(n+k-1)! ) for k=5.

(n+4)*(n+3)*(n+2)*(n+1)*a(n) -5*(5*n-4)*(5*n-3)*(5*n-2)*(5*n-1)*a(n-1)=0. - R. J. Mathar, Aug 10 2015

G.f.: x*5F4(1,6/5,7/5,8/5,9/5;3,4,5,6;3125*x). - R. J. Mathar, Aug 10 2015

a(n) ~ 72*5^(5*n+1/2)/(Pi^2*n^12). - Vaclav Kotesovec, Nov 18 2016

E.g.f.: 4F4(1/5,2/5,3/5,4/5; 2,3,4,5; 3125*x). - Ilya Gutkovskiy, Oct 13 2017

Maple

a:= n-> (5*n)! * mul(i!/(n+i)!, i=0..4):
seq(a(n), n=0..20);  # Alois P. Heinz, Jul 23 2017

Mathematica

Table[288*(5*n)!/(n!*(n+1)!*(n+2)!*(n+3)!*(n+4)!), {n, 1, 20}] (* Vaclav Kotesovec, Nov 18 2016 *)

Crossrefs

A row of A060854.

Sequence in context: A091545 A101630 A273628 * A167668 A277665 A215837

Adjacent sequences: A005788 A005789 A005790 * A005792 A005793 A005794

Keyword

nonn,easy

Author

N. J. A. Sloane.

Extensions

a(0)=1 prepended by Alois P. Heinz, Jul 23 2017

Status

approved