A274308 -id:A274308 - cp's OEIS frontend

A284308 Number A(n,k) of singular vector tuples for a general k-dimensional {n}^k tensor; square array A(n,k), n>=1, k>=1, read by antidiagonals.

1, 1, 1, 1, 2, 1, 1, 6, 3, 1, 1, 24, 37, 4, 1, 1, 120, 997, 240, 5, 1, 1, 720, 44121, 51264, 1621, 6, 1, 1, 5040, 2882071, 23096640, 2940841, 11256, 7, 1, 1, 40320, 260415373, 18754813440, 14346274601, 180296088, 79717, 8, 1, 1, 362880, 31088448777, 24874143759360, 153480509680141, 9859397817600, 11559133741, 572928, 9, 1

Offset: 1

Alois P. Heinz, Mar 24 2017

			Square array A(n,k) begins:
  1, 1,     1,         1,             1,                   1, ...
  1, 2,     6,        24,           120,                 720, ...
  1, 3,    37,       997,         44121,             2882071, ...
  1, 4,   240,     51264,      23096640,         18754813440, ...
  1, 5,  1621,   2940841,   14346274601,     153480509680141, ...
  1, 6, 11256, 180296088, 9859397817600, 1435747717722810960, ...

Alois P. Heinz, Antidiagonals n = 1..18, flattened
Shalosh B. Ekhad and Doron Zeilberger, On the Number of Singular Vector Tuples of Hyper-Cubical Tensors, 2016; also arXiv preprint arXiv:1605.00172, 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.
Bernd Sturmfels, Tensors and Their Eigenvalues, Notices AMS, 63 (No. 6, 2016), 606-606.

Columns k=1-9 give: A000012, A000027, A271905, A272551, A283829, A283830, A287083, A287084, A287085.
Rows n=1-3 give: A000012, A000142, A274308.
Main diagonal gives A284309.

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

Original entry on oeis.org

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

Doron Zeilberger, Apr 21 2016

nonn

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

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.

See A272551 for the n X n X n X n version.
Column k=3 of A284308.
Cf. A274308.

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 *)

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.

A321711 Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section.

Original entry on oeis.org

1, 1, 0, 3, 0, 0, 11, 9, 0, 1, 53, 120, 60, 40, 9, 309, 1410, 1800, 1590, 885, 216, 2119, 16560, 39960, 55120, 52065, 29016, 7570, 16687, 202755, 801780, 1696555, 2433165, 2300403, 1326850, 357435, 148329, 2624496, 15606360, 48387024, 99650670, 141429456, 135382464, 79738800, 22040361, 1468457, 36080100, 304274880, 1323453180, 3760709526, 7493549868, 10570597800, 10199809980, 6103007505, 1721632024

Offset: 0

Gheorghe Coserea, Nov 27 2018

			For n=3 we have s1 = z1 + z2 + z3, s2 = z1^2 + z2^2 + z3^2, s12 = z1*z2 + z1*z3 + z2*z3, f1 = z1^2 + z2^2 + z3^2 + t*z2*z3 + z1*(z2 + z3), f2 = z1^2 + z2^2 + z3^2 + t*z1*z3 + z2*(z1 + z3), f3 = z1^2 + z2^2 + z3^2 + t*z1*z2 + z3*(z1 + z2), [(z1*z2*z3)^2] f1*f2*f3 = 11 + 9*t + t^3, therefore P_3(t) = 11 + 9*t + t^3.
A(x;t) = 1 + x + 3*x^2 + (11 + 9*t + t^3)*x^3 + (53 + 120*t + 60*t^2 + 40*t^3 + 9*t^4)*x^4 + ...
Triangle starts:
n\k [0]    [1]     [2]     [3]      [4]      [5]      [6]      [7]
[0] 1;
[1] 1;     0;
[2] 3;     0;      0;
[3] 11,    9,      0,      1;
[4] 53,    120,    60,     40,      9;
[5] 309,   1410,   1800,   1590,    885,     216;
[6] 2119,  16560,  39960,  55120,   52065,   29016,   7570;
[7] 16687, 202755, 801780, 1696555, 2433165, 2300403, 1326850, 357435;
[8] ...

Gheorghe Coserea, Rows n = 0..13, flattened
Shmuel Friedland, Giorgio Ottaviani, The number of singular vector tuples and uniqueness of best rank one approximation of tensors, arXiv:1210.8316 [math.AG], 2013.

Cf. A000255, A000681, A007107, A274308, A284989.

P(n, t='t) = {
  my(z=vector(n, k, eval(Str("z", k))),
     s1=sum(k=1, #z, z[k]), s2=sum(k=1, #z, z[k]^2), s12=(s1^2 - s2)/2,
     f=vector(n, k, s2 + t*(s12 - z[k]*(s1 - z[k])) + z[k]*(s1 - z[k])), g=1);
  for (i=1, n, g *= f[i]; for(j=1, n, g=substpol(g, z[j]^3, 0)));
  for (k=1, n, g=polcoef(g, 2, z[k]));
  g;
};
seq(N) = concat([[1], [1, 0], [3, 0, 0]], apply(n->Vecrev(P(n,'t)), [3..N]));
concat(seq(9))

Let z1..zn be n variables and s1 = Sum_{k=1..n} zk, s2 = Sum_{k=1..n} zk^2, s12 = (s1^2 - s2)/2, fk = s2 + t*(s12 - zk*(s1 - zk)) + zk*(s1 - zk) for k=1..n; we define P_n(t) = [(z1..zn)^2] Product_{k=1..n} fk.
A000255(n) = T(n,0).
A007107(n) = T(n,n).
A000681(n) = Sum_{k=0..n} T(n,k).
A274308(n) = Sum_{k=0..n} T(n,k)*2^k.

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A284308 Number A(n,k) of singular vector tuples for a general k-dimensional {n}^k tensor; square array A(n,k), n>=1, k>=1, read by antidiagonals.

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

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A321711 Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section.

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