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

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

Offset

1,2

Links

Table of n, a(n) for n=1..16.

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.)

Maple

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;

Mathematica

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 *)

Prog

(PARI)
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

Crossrefs

Row n=3 of A284308.

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

Sequence in context: A003717 A354020 A201697 * A318224 A300986 A003716

Adjacent sequences: A274305 A274306 A274307 * A274309 A274310 A274311

Keyword

nonn

Author

N. J. A. Sloane, Jun 21 2016

Extensions

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

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

Status

approved