A284308 -id:A284308 - cp's OEIS frontend

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

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.

A272551 Number of singular vector tuples for a general 4-dimensional n X n X n X n tensor.

Original entry on oeis.org

1, 24, 997, 51264, 2940841, 180296088, 11559133741, 765337680384, 51921457661905, 3590122671128664, 252070718210663749, 17922684123178825536, 1287832671004683373753, 93368940577497932331288, 6821632357294515590873917, 501741975445243527381995520, 37121266623211130111114816929

Offset: 1

Doron Zeilberger, May 02 2016

nonn

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.

See A271905 for the three-dimensional analog.

Column k=4 of A284308.

a[n_] := Module[{a, b, c, d, s}, s = Series[(
  ((a + b + c)^n - d^n)*
  ((b + c + d)^n - a^n)*
  ((c + d + a)^n - b^n)*
  ((d + a + b)^n - c^n))/(
  (a + b + c - d)*
  (b + c + d - a)*
  (c + d + a - b)*
  (d + a + b - c)),
  {a, 0, n}, {b, 0, n}, {c, 0, n}, {d, 0, n}] // Normal // Expand;
  Cases[List @@ s, k_Integer a^(n-1) b^(n-1) c^(n-1) d^(n-1)] /. (a|b|c|d) -> 1 // First
];
Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 1, 17}] (* Jean-François Alcover, Aug 19 2018, after A271905 *)

A283830 Number of singular vector tuples for a general 6-dimensional n X n X n X n X n X n tensor.

Original entry on oeis.org

1, 720, 2882071, 18754813440, 153480509680141, 1435747717722810960, 14662732377776152127011, 159330378168761744514908160, 1813222281365129112322849988761, 21386175398690803114094640652896720, 259532087509984537826921145014495182351

Offset: 1

N. J. A. Sloane, Mar 19 2017

nonn

Shalosh B. Ekhad and Doron Zeilberger, On the Number of Singular Vector Tuples of Hyper-Cubical Tensors, arXiv preprint arXiv:1605.00172, 2016. [This is a different document from the one with the same title on Doron Zeilberger's web site]

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 [Local copy, pdf file only, no active links]

Cf. A271905, A272551, A283829.

Column k=6 of A284308.

More terms from Alois P. Heinz, Mar 24 2017

A274308 Number of n-tuples of singular vectors of a 3 X 3 X 3 X ... X 3 n-dimensional tensor.

Original entry on oeis.org

1, 3, 37, 997, 44121, 2882071, 260415373, 31088448777, 4737782756017, 897380763253291, 206773800208348341, 56951114596754707693, 18476855531112777659017, 6973886287904020598308287, 3029760395576715276955711261, 1501087423496953812426438796561

Offset: 1

N. J. A. Sloane, Jun 21 2016

nonn

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.
Bernd Sturmfels, Tensors and Their Eigenvalues, Notices AMS, 63 (No. 6, 2016), 606-606. (Th. 9 gives g.f.)

Row n=3 of A284308.

Cf. A271905, A272551, A283829, A283830, A321711.

ans:=[];
for d from 1 to 10 do
for h from 1 to d do zh[h]:=add(z[i],i=1..d)-z[h]; od;
t1:= expand(simplify( mul( (zh[i]^3-z[i]^3) / (zh[i]-z[i]), i=1..d)));
a:=t1; for i from 1 to d do a:=coeff(a,z[i],2); od;
ans:=[op(ans),a];
od:
ans;

a[n_] := Module[{s, x, xx, xd, f}, s = Total[xx = Array[x, n]]; xd = {#, 0, 2}& /@ xx; f = 1; Do[f = Series[f(s^2 - s x[i] + x[i]^2), Sequence @@ Evaluate[xd]], {i, 1, n}]; SeriesCoefficient[f, Sequence @@ Evaluate[xd]] ];
Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 1, 12}] (* Jean-François Alcover, Nov 26 2018 *)

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;
};
vector(10, n, P(n,2)) \\ Gheorghe Coserea, Nov 27 2018

a(11)-a(15) from Gheorghe Coserea, Jun 29 2016

a(16) from Alois P. Heinz, Mar 24 2017

A283829 Number of singular vector tuples for a general 5-dimensional n X n X n X n X n tensor.

Original entry on oeis.org

1, 120, 44121, 23096640, 14346274601, 9859397817600, 7244702262723241, 5582882474985676800, 4458184170040912112721, 3659860624802972467991520, 3071619882576430968562820921, 2624927569719125879186684147520, 2277214648583912948194825667564881

Offset: 1

N. J. A. Sloane, Mar 19 2017

nonn

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.

Cf. A271905, A272551, A283830.

Column k=5 of A284308.

More terms from Alois P. Heinz, Mar 24 2017

A284309 Number of singular vector tuples for a general n-dimensional {n}^n tensor.

Original entry on oeis.org

1, 2, 37, 51264, 14346274601, 1435747717722810960, 79118094349714452632485774477, 3409699209687052091502059492845005192560640, 154730604283618051465998344012575355916858352712971348277665, 9576184829775011641104888042379740657096306109466956243538100418643876547244800

Offset: 1

Alois P. Heinz, Mar 24 2017

nonn

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.

Main diagonal of A284308.

a[1] = 1;
a[n_] := Module[{Z, z, P},
  Z[i_] := Sum[z[k], {k, 1, n}] - z[i];
  P = Product[(Z[i]^n - z[i]^n)/(Z[i] - z[i]), {i, 1, n}] // Cancel;
  SeriesCoefficient[P, Sequence @@ Table[{z[i], 0, n-1}, {i, 1, n}]]
];
Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 1, 5}] (* Jean-François Alcover, Aug 06 2018 *)

A287083 Number of singular vector tuples for a general 7-dimensional {n}^7 tensor.

Original entry on oeis.org

1, 5040, 260415373, 24874143759360, 3160621587907016605, 473787689225018384094480, 79118094349714452632485774477, 14256246398333360805158745610321920, 2718180337074137745147914900104417897849, 541385910515057381740441349336869933006303200

Offset: 1

Alois P. Heinz, May 19 2017

nonn

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.

Column k=7 of A284308.

A287084 Number of singular vector tuples for a general 8-dimensional {n}^8 tensor.

Original entry on oeis.org

1, 40320, 31088448777, 50257106410291200, 114196733016257979480401, 315745472497043733378660720000, 992004278721114611076713896739874457, 3409699209687052091502059492845005192560640, 12527584332978774512983761712826111596464807512481

Offset: 1

Alois P. Heinz, May 19 2017

nonn

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.

Column k=8 of A284308.

A287085 Number of singular vector tuples for a general 9-dimensional {n}^9 tensor.

Original entry on oeis.org

1, 362880, 4737782756017, 146830181787844853760, 6753239838803803486306487761, 389685915793550955923351453734807680, 26057203877175385471624399463506301352492241, 1932393393532020981741986795273501624377597577134080

Offset: 1

Alois P. Heinz, May 19 2017

nonn

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.

Column k=9 of A284308.

cp's OEIS Frontend

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

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A272551 Number of singular vector tuples for a general 4-dimensional n X n X n X n tensor.

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A283830 Number of singular vector tuples for a general 6-dimensional n X n X n X n X n X n tensor.

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A274308 Number of n-tuples of singular vectors of a 3 X 3 X 3 X ... X 3 n-dimensional tensor.

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A283829 Number of singular vector tuples for a general 5-dimensional n X n X n X n X n tensor.

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A284309 Number of singular vector tuples for a general n-dimensional {n}^n tensor.

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Mathematica

A287083 Number of singular vector tuples for a general 7-dimensional {n}^7 tensor.

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A287084 Number of singular vector tuples for a general 8-dimensional {n}^8 tensor.

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A287085 Number of singular vector tuples for a general 9-dimensional {n}^9 tensor.

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