login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A271905 Number of singular vector tuples for a general n X n X n tensor. 6

%I

%S 1,6,37,240,1621,11256,79717,572928,4164841,30553116,225817021,

%T 1679454816,12556853401,94313192616,711189994357,5381592930816,

%U 40848410792017,310909645663332,2372280474687277,18141232682656320,139010366280363601,1067160872528170536,8206301850166625797,63203453697218605440

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

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

%H Alois P. Heinz, <a href="/A271905/b271905.txt">Table of n, a(n) for n = 1..1111</a>

%H Shalosh B. Ekhad and Doron Zeilberger, <a href="http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/svt.html">On the Number of Singular Vector Tuples of Hyper-Cubical Tensors</a>, 2016.

%H Shalosh B. Ekhad and Doron Zeilberger, <a href="https://arxiv.org/abs/1605.00172">On the number of Singular Vector Tuples of Hyper-Cubical Tensors</a>, arXiv preprint arXiv:1605.00172 [math.CO], 2016.

%H Shmuel Friedland and Giorgio Ottaviani, <a href="http://dx.doi.org/10.1007/s10208-014-9194-z">The number of singular vector tuples and uniqueness of best rank-one approximation of tensors</a>, Found. Comput. Math. 14 (2014), no. 6, 1209-1242.

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

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

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

%F 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

%F with initial conditions

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

%F and asymptotically

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

%t a[1] = 1;

%t 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];

%t Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 1, 24}] (* _Jean-Fran├žois Alcover_, Aug 18 2018 *)

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

%Y Column k=3 of A284308.

%Y Cf. A274308.

%K nonn

%O 1,2

%A _Doron Zeilberger_, Apr 21 2016

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified July 29 22:13 EDT 2021. Contains 346346 sequences. (Running on oeis4.)