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A059486
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3-enumeration of 2n+1 X 2n+1 vertically symmetric alternating-sign matrices.
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4
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1, 1, 5, 126, 16038, 10320453, 33590259846, 553104735325740, 46084184498427053436, 19430969437346561065941390, 41463730793298298041665385308325, 447814224393522724673729884056814834500, 24479424309393636290695101063892553945412075000
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OFFSET
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0,3
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LINKS
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FORMULA
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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
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MAPLE
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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
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MATHEMATICA
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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[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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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