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A059477 3-enumeration of n X n alternating-sign matrices. 3
1, 1, 2, 9, 90, 2025, 102060, 11573604, 2946308904, 1687603650084, 2171945897658108, 6289412333143466241, 40940643700218614247324, 599627833263501883888374756, 19747212169938041691404746667280, 1463229065460461810019231236067824400 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

REFERENCES

Paul Barry, Jacobsthal Decompositions of Pascal's Triangle, Ternary Trees, and Alternating Sign Matrices, Journal of Integer Sequences, 19, 2016, #16.3.5.

LINKS

Table of n, a(n) for n=0..15.

G. Kuperberg, Symmetry classes of alternating-sign matrices under one roof, arXiv math.CO/0008184

F. Colomo and A. G. Pronko, On the refined 3-enumeration of alternating sign matrices.

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, JSTAT (2005) P01005.

Yu. G. Stroganov, 3-enumerated alternating sign matrices, math-ph/0304004.

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

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;

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-Fran├žois Alcover, Dec 28 2017, after Ralf Stephan *)

CROSSREFS

Cf. A005130.

Sequence in context: A006120 A012941 A216691 * A136553 A266293 A224323

Adjacent sequences:  A059474 A059475 A059476 * A059478 A059479 A059480

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Feb 04 2001

STATUS

approved

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Last modified August 18 17:55 EDT 2018. Contains 313834 sequences. (Running on oeis4.)