A059477 3-enumeration of n X n alternating-sign matrices.
1, 1, 2, 9, 90, 2025, 102060, 11573604, 2946308904, 1687603650084, 2171945897658108, 6289412333143466241, 40940643700218614247324, 599627833263501883888374756, 19747212169938041691404746667280, 1463229065460461810019231236067824400
Offset: 0
Keywords
Links
- Paul Barry, Jacobsthal Decompositions of Pascal's Triangle, Ternary Trees, and Alternating Sign Matrices, Journal of Integer Sequences, 19, 2016, #16.3.5.
- F. Colomo and A. G. Pronko, On the refined 3-enumeration of alternating sign matrices, arXiv:math-ph/0404045, 2004.
- F. Colomo and A. G. Pronko, On the refined 3-enumeration of alternating sign matrices, Advances in Applied Mathematics 34 (2005) 798.
- F. Colomo and A. G. Pronko, Square ice, alternating sign matrices and classical orthogonal polynomials, arXiv:math-ph/0411076, 2004; JSTAT (2005) P01005.
- G. Kuperberg, Symmetry classes of alternating-sign matrices under one roof, arXiv:math/0008184 [math.CO], 2000-2001.
- Yu. G. Stroganov, 3-enumerated alternating sign matrices, arXiv:math-ph/0304004, 2003.
Crossrefs
Cf. A005130.
Programs
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Maple
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;
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Mathematica
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}]; a[n_?EvenQ] := (3^((n-2)/2)*((3*(n-2))/2 + 2)!*((n - 2)/2)! * a[n - 1])/(n - 1)!^2; Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Dec 28 2017, after Ralf Stephan *)
Formula
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