A283829 -id:A283829 - 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.

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

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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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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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