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A005790 4-dimensional Catalan numbers.

%I A005790 M4954 #74 Aug 27 2025 11:39:34
%S A005790 1,1,14,462,24024,1662804,140229804,13672405890,1489877926680,
%T A005790 177295473274920,22661585038594320,3073259571003214320,
%U A005790 438091463242348309440,65166105157299311029200,10056663345892631910888600,1602608179958939072505281850,262708662267696303439658400600
%N A005790 4-dimensional Catalan numbers.
%C A005790 Number of standard tableaux of shape (n,n,n,n). - _Emeric Deutsch_, May 13 2004
%C A005790 The prime terms (as defined in A268538) are 1, 1, 10, 320, 16764, 1171355, 99315236, 9691755128, 1053114415100, ... - _R. J. Mathar_, Feb 27 2018
%D A005790 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D A005790 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.
%H A005790 Seiichi Manyama, <a href="/A005790/b005790.txt">Table of n, a(n) for n = 0..423</a> (terms 1..130 from Alois P. Heinz)
%H A005790 Shalosh B. Ekhad and Doron Zeilberger, <a href="http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/ssyt.html">Computational and Theoretical Challenges on Counting Solid Standard Young Tableaux</a>. Also arXiv preprint arXiv:1202.6229, 2012. - _N. J. A. Sloane_, Jul 07 2012
%H A005790 Michaël Moortgat, <a href="https://cla.tcs.uj.edu.pl/history/2020/pdfs/CLA_Moortgat.pdf">The Tamari order for D^3 and derivability in semi-associative Lambek-Grishin Calculus</a>, 15th Workshop: Computational Logic and Applications (CLA 2020).
%H A005790 Katarzyna Górska and Karol A. Penson, <a href="http://arxiv.org/abs/1304.6008">Multidimensional Catalan and related numbers as Hausdorff moments</a>, arXiv preprint arXiv:1304.6008 [math.CO], 2013.
%H A005790 Dimana Miroslavova Pramatarova, <a href="https://www.cee.org/sites/default/files/rsi/Papers/pramatarovadimana_193982_5452885_Pramatarova_Dimana_AnaMaria_Final.pdf">Investigating the Periodicity of Weighted Catalan Numbers and Generalizing Them to Higher Dimensions</a>, MIT Res. Sci. Instit. (2025). See p. 13.
%H A005790 Stephen Snover, <a href="/A005789/a005789.pdf">Letter to N. J. A. Sloane, May 1991</a>
%H A005790 Stephanie F. Troyer and Stephen L. Snover, <a href="/A005789/a005789_1.pdf">m-Dimensional Catalan numbers</a>, Preprint, 1989. (Annotated scanned copy)
%F A005790 a(n) = 12*(4*n)!/(n! *(n+1)! *(n+2)! *(n+3)!).
%F A005790 G.f.: 4_F_3 ( [ 1, 3/2, 5/4, 7/4 ]; [ 3, 4, 5 ]; 256 x ).
%F A005790 a(n) ~ 3*2^(8*n+3/2)/(Pi^(3/2)*n^(15/2)). - _Vaclav Kotesovec_, Nov 18 2016
%F A005790 E.g.f.: 3F3(1/4,1/2,3/4; 2,3,4; 256*x) - 1. - _Ilya Gutkovskiy_, Oct 13 2017
%F A005790 (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
%p A005790 a:= n-> (4*n)! * mul(i!/(4+i)!, i=0..n-1):
%p A005790 seq(a(n), n=0..20);  # _Alois P. Heinz_, Jul 25 2012
%t A005790 Table[12*(4*n)!/(n!*(n+1)!*(n+2)!*(n+3)!), {n, 0, 20}] (* _Vaclav Kotesovec_, Nov 18 2016 *)
%o A005790 (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
%o A005790 (PARI) vector(20, n, n--; 12*(4*n)!/(n!*(n+1)!*(n+2)!*(n+3)!)) \\ _G. C. Greubel_, Nov 23 2018
%o A005790 (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
%Y A005790 A row of A060854.
%Y A005790 Cf. A000108 (Catalan numbers), A005789, A005791.
%K A005790 nonn,easy,changed
%O A005790 0,3
%A A005790 _N. J. A. Sloane_
%E A005790 a(0)=1 prepended by _Seiichi Manyama_, Nov 23 2018