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A059477 3-enumeration of n X n alternating-sign matrices.

%I A059477 #20 Jan 29 2019 10:07:13
%S A059477 1,1,2,9,90,2025,102060,11573604,2946308904,1687603650084,
%T A059477 2171945897658108,6289412333143466241,40940643700218614247324,
%U A059477 599627833263501883888374756,19747212169938041691404746667280,1463229065460461810019231236067824400
%N A059477 3-enumeration of n X n alternating-sign matrices.
%H A059477 Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Barry/barry321.html">Jacobsthal Decompositions of Pascal's Triangle, Ternary Trees, and Alternating Sign Matrices</a>, Journal of Integer Sequences, 19, 2016, #16.3.5.
%H A059477 F. Colomo and A. G. Pronko, <a href="https://arxiv.org/abs/math-ph/0404045">On the refined 3-enumeration of alternating sign matrices</a>, arXiv:math-ph/0404045, 2004.
%H A059477 F. Colomo and A. G. Pronko, <a href="https://doi.org/10.1016/j.aam.2004.09.007">On the refined 3-enumeration of alternating sign matrices</a>, Advances in Applied Mathematics 34 (2005) 798.
%H A059477 F. Colomo and A. G. Pronko, <a href="http://www.arXiv.org/abs/math-ph/0411076">Square ice, alternating sign matrices and classical orthogonal polynomials</a>, arXiv:math-ph/0411076, 2004; JSTAT (2005) P01005.
%H A059477 G. Kuperberg, <a href="https://arxiv.org/abs/math/0008184">Symmetry classes of alternating-sign matrices under one roof</a>, arXiv:math/0008184 [math.CO], 2000-2001.
%H A059477 Yu. G. Stroganov, <a href="https://arxiv.org/abs/math-ph/0304004">3-enumerated alternating sign matrices</a>, arXiv:math-ph/0304004, 2003.
%F A059477 a(2m+1)=3^(m*(m+1))*prod(k=1, m, ((3*k-1)!/(m+k)!)^2), a(2m+2)=3^m*(3*m+2)!*m!/((2*m+1)!)^2*a(2m+1). - _Ralf Stephan_, Apr 24 2004
%p A059477 A059477 := proc(n) local i,j,t1; t1 := 3^(n^2-n)*2^(-n^2+n); for i from 1 to n do for j from 1 to n do if j-i mod 2 <> 0 then t1 := t1*(3*j-3*i+1)/(3*j-3*i); fi; od; od; t1; end;
%t A059477 a[0] = 1; a[n_?OddQ] := a[n] = 3^((1/2)*((n-1)/2 + 1)*(n-1)) * Product[(3*k - 1)!^2/(k + (n-1)/2)!^2, {k, 1, (n - 1)/2}];
%t A059477 a[n_?EvenQ] := (3^((n-2)/2)*((3*(n-2))/2 + 2)!*((n - 2)/2)! * a[n - 1])/(n - 1)!^2;
%t A059477 Table[a[n], {n, 0, 15}] (* _Jean-François Alcover_, Dec 28 2017, after _Ralf Stephan_ *)
%Y A059477 Cf. A005130.
%K A059477 nonn
%O A059477 0,3
%A A059477 _N. J. A. Sloane_, Feb 04 2001