cp's OEIS Frontend

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

%I A271905 #40 Aug 18 2018 16:43:13
%S A271905 1,6,37,240,1621,11256,79717,572928,4164841,30553116,225817021,
%T A271905 1679454816,12556853401,94313192616,711189994357,5381592930816,
%U A271905 40848410792017,310909645663332,2372280474687277,18141232682656320,139010366280363601,1067160872528170536,8206301850166625797,63203453697218605440
%N A271905 Number of singular vector tuples for a general n X n X n tensor.
%D A271905 Bernd Sturmfels, Eigenvectors of Tensors, Colloquium Talk, Rutgers University, Apr 22 2016.
%H A271905 Alois P. Heinz, <a href="/A271905/b271905.txt">Table of n, a(n) for n = 1..1111</a>
%H A271905 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 A271905 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 A271905 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 A271905 From Eq. (1.3) of Ottaviani-Friedland (2014), a(n) is the coefficient of (abc)^(n-1) in the polynomial
%F A271905 {((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 A271905 a(n) satisfies the following fifth-order recurrence equation with polynomial coefficients:
%F A271905 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 A271905 with initial conditions
%F A271905   [a(1), ..., a(5)] = [1, 6, 37, 240, 1621]
%F A271905 and asymptotically
%F A271905   a(n) ~ (2/(sqrt(3)*Pi))*8^n/n.
%t A271905 a[1] = 1;
%t A271905 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 A271905 Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 1, 24}] (* _Jean-François Alcover_, Aug 18 2018 *)
%Y A271905 See A272551 for the n X n X n X n version.
%Y A271905 Column k=3 of A284308.
%Y A271905 Cf. A274308.
%K A271905 nonn
%O A271905 1,2
%A A271905 _Doron Zeilberger_, Apr 21 2016