A059486 3-enumeration of 2n+1 X 2n+1 vertically symmetric alternating-sign matrices.

1, 1, 5, 126, 16038, 10320453, 33590259846, 553104735325740, 46084184498427053436, 19430969437346561065941390, 41463730793298298041665385308325, 447814224393522724673729884056814834500, 24479424309393636290695101063892553945412075000

Offset: 0

Author

N. J. A. Sloane, Feb 04 2001

Keywords

• nonn
• easy

Links

• Harry J. Smith, Table of n, a(n) for n = 0..53
• G. Kuperberg, Symmetry classes of alternating-sign matrices under one roof, arXiv:math/0008184 [math.CO], 2000-2001. [Th. 3, but the formula there is incorrect]
• J. Propp, The many faces of alternating-sign matrices, Discrete Mathematics and Theoretical Computer Science Proceedings AA (DM-CCG), 2001, 43-58.

Crossrefs

Cf. A025748, A227379.

Programs

Maple

A059486 := proc(n) local i, j, t1; t1 := 3^(2*n^2)/2^(2*n^2 + n); for i to 2*n + 1 do for j to 2*n + 1 do if i mod 2 <> 0 and j mod 2 = 0 then t1 := t1*(3*j - 3*i + 1)/(3*j - 3*i) end if end do end do; t1 end proc;
e(n)= { local(A); A=Vec((1 - (1 - 9*x + O(x^(2*n + 1)))^(1/3))/(3*x)); matdet(matrix(n, n, i, j, A[i+j]))/3^n; } { for (n = 0, 100, a=e(n); if (a > 10^(10^3 - 6), break); write("b059486.txt", n, " ", a); ) } # Harry J. Smith, Jun 27 2009

Mathematica

a[n_] := Module[{i, j, t1}, t1 = 3^(2*n^2)/2^(2*n^2 + n); For[i = 1, i <= 2*n + 1, i++, For[j = 1, j <= 2*n + 1, j++, If[Mod[i, 2] != 0 && Mod[j, 2] == 0, t1 = t1*(3*j - 3*i + 1)/(3*j - 3*i)]]]; t1];
Table[a[n], {n, 0, 12}] (* Jean-François Alcover, Nov 23 2017, translated from Maple *)
Table[3^(2*n^2)/2^(2*n^2 + n) * Product[(2 + 6*i - 6*j)/(3 + 6*i - 6*j), {i, 0, n}, {j, 1, n}], {n, 0, 15}] (* Vaclav Kotesovec, Feb 24 2019 *)

PARI

a(n)=local(A); if(n<0,0,A=Vec((1-(1-9*x+O(x^(2*n+1)))^(1/3))/(3*x)); matdet(matrix(n,n,i,j,A[i+j]))/3^n)

PARI

e(n)= { local(A); A=Vec((1 - (1 - 9*x + O(x^(2*n + 1)))^(1/3))/(3*x)); matdet(matrix(n, n, i, j, A[i+j]))/3^n; } { for (n = 0, 100, a=e(n); if (a > 10^(10^3 - 6), break); write("b059486.txt", n, " ", a); ) } \\ Harry J. Smith, Jun 27 2009

Formula

a(n) ~ exp(1/36) * Gamma(1/3)^(1/3) * 3^(n*(4*n + 1)/2 + 11/36) * n^(1/36) / (2^(2*n*(n+1) + 7/12) * A^(1/3) * Pi^(1/6)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Feb 24 2019