This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A123555 #27 Dec 30 2023 12:09:59
%S A123555 0,2,16,168,2112,30030,466752,7759752,135980416,2485891980,
%T A123555 47052314400,916847954880,18311313000960,373542610526280,
%U A123555 7761573156274560,163893933165976200,3510476121410184960,76151734612882397700,1670824967127762045600,37036620104665392010800,828632324276985756528000
%N A123555 Number of standard Young tableaux of type (n+1,n,n-1).
%C A123555 For n > 0, a(n) is the number of up-down permutations of length 2n + 1 with no four-term increasing subsequence. Equivalently, this is the number of up-down permutations of length 2n + 1 with no four-term decreasing subsequence; the number of down-up permutations of length 2n + 1 with no four-term increasing subsequence; and the number of down-up permutations of length 2n + 1 with no four-term decreasing subsequence. (An up-down permutation is one whose descent set is {2, 4, 6, ...}.). - _Joel B. Lewis_, Oct 05 2009
%D A123555 For definition see James and Kerber, Representation Theory of Symmetric Group, Addison-Wesley, 1981, p. 107.
%H A123555 G. C. Greubel, <a href="/A123555/b123555.txt">Table of n, a(n) for n = 0..700</a>
%H A123555 Joerg Arndt, <a href="/A123555/a123555.txt">The a(3)=168 Young tableaux of shape [4,3,2]</a>.
%H A123555 Joel B. Lewis, <a href="http://arxiv.org/abs/0909.4966">Pattern avoidance and RSK-like algorithms for alternating permutations and Young tableaux</a>, arXiv:0909.4966 [math.CO], 2009-2011. [_Joel B. Lewis_, Oct 05 2009]
%H A123555 Joel B. Lewis, <a href="https://dspace.mit.edu/handle/1721.1/73444">Pattern Avoidance for Alternating Permutations and Reading Words of Tableaux</a>, Ph. D. Dissertation, Department of Mathematics, MIT, 2012.
%H A123555 Sherry H. F. Yan, <a href="https://doi.org/10.37236/3243">On Wilf equivalence for alternating permutations</a>, Elect. J. Combinat.; 20 (2013), #P58.
%H A123555 <a href="/index/Y#Young">Index entries for sequences related to Young tableaux.</a>
%F A123555 a(n) = 16*(3*n)!/((n-1)!*(n+1)!*(n+3)!).
%F A123555 (n-1)*(n+3)*(n+1)*a(n) -3*n*(3*n-1)*(3*n-2)*a(n-1)=0, n>1. - _R. J. Mathar_, Aug 10 2015
%F A123555 G.f.: 2x*3F2(5/3,4/3,2;3,5;27x). - _R. J. Mathar_, Aug 10 2015
%t A123555 f[n_]:=16 (3 n)!/((n-1)! (n+1)! (n+3)!)
%t A123555 (* first do *) Needs["DiscreteMath`Combinatorica`"] (* then *) Table[ NumberOfTableaux@ {n + 1, n, n - 1}, {n, 0, 17}] (* _Robert G. Wilson v_ *)
%o A123555 (PARI) for(n=0,25, print1(16*(3*n)!/((n-1)!*(n+1)!*(n+3)!), ", ")) \\ _G. C. Greubel_, Oct 15 2017
%Y A123555 Cf. A011553.
%Y A123555 For up-down permutations of even length, see A005789. [_Joel B. Lewis_, Oct 05 2009]
%K A123555 nonn,easy
%O A123555 0,2
%A A123555 Amitai Regev (amitai.regev(AT)weizmann.ac.il), Nov 15 2006