A271905 Number of singular vector tuples for a general n X n X n tensor.

1, 6, 37, 240, 1621, 11256, 79717, 572928, 4164841, 30553116, 225817021, 1679454816, 12556853401, 94313192616, 711189994357, 5381592930816, 40848410792017, 310909645663332, 2372280474687277, 18141232682656320, 139010366280363601, 1067160872528170536, 8206301850166625797, 63203453697218605440

Offset: 1

Author

Doron Zeilberger, Apr 21 2016

Keywords

• nonn

References

• Bernd Sturmfels, Eigenvectors of Tensors, Colloquium Talk, Rutgers University, Apr 22 2016.

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..1111
• Shalosh B. Ekhad and Doron Zeilberger, On the Number of Singular Vector Tuples of Hyper-Cubical Tensors, 2016.
• Shalosh B. Ekhad and Doron Zeilberger, On the number of Singular Vector Tuples of Hyper-Cubical Tensors, arXiv preprint arXiv:1605.00172 [math.CO], 2016.
• Shmuel Friedland and Giorgio Ottaviani, The number of singular vector tuples and uniqueness of best rank-one approximation of tensors, Found. Comput. Math. 14 (2014), no. 6, 1209-1242.

Crossrefs

See A272551 for the n X n X n X n version.

Column k=3 of A284308.

Cf. A274308.

Programs

Mathematica

a[1] = 1;
a[n_] := Module[{a, b, c, s}, s = Series[(((a + b)^n - c^n)((a + c)^n - b^n)((b + c)^n - a^n))/((a + b - c)(a + c - b)(b + c - a)), {a, 0, n}, {b, 0, n}, {c, 0, n}] // Normal // Expand; Cases[List @@ s, k_Integer a^(n-1) b^(n-1) c^(n-1)] /. (a|b|c) -> 1 // First];
Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 1, 24}] (* Jean-François Alcover, Aug 18 2018 *)

Formula

From Eq. (1.3) of Ottaviani-Friedland (2014), a(n) is the coefficient of (abc)^(n-1) in the polynomial

{((a+b)^n-c^n)*((a+c)^n-b^n)*((b+c)^n-a^n)} / {(a+b-c)*(a+c-b)*(b+c-a)}.

a(n) satisfies the following fifth-order recurrence equation with polynomial coefficients:

72*(n + 2)*(245*n^4 + 3094*n^3 + 14447*n^2 + 29474*n + 22100)*(n + 1)^2*a(n) - (n + 2)*(21805*n^6 + 330981*n^5 + 2012733*n^4 + 6230951*n^3 + 10263446*n^2 + 8425060*n + 2639760)*a(n + 1) + (-10279296 - 13230*n^7 - 249641*n^6 - 29331496*n - 22847777*n^3 - 1998705*n^5 - 8785333*n^4 - 35069178*n^2)*a(n + 2) + (16026528 + 21560*n^7 + 413637*n^6 + 3343917*n^5 + 14735333*n^4 + 38132651*n^3 + 57777574*n^2 + 47273504*n)*a(n + 3) - (4410*n^6 + 70147*n^5 + 452903*n^4 + 1516515*n^3 + 2769127*n^2 + 2601986*n + 975888)*(n + 4)*a(n + 4) + (n + 5)*(n + 4)*(n + 3)*(245*n^4 + 2114*n^3 + 6635*n^2 + 8882*n + 4224)*a(n + 5) = 0

with initial conditions

[a(1), ..., a(5)] = [1, 6, 37, 240, 1621]

and asymptotically

a(n) ~ (2/(sqrt(3)*Pi))*8^n/n.