A274308 - OEIS

%I #33 Dec 01 2018 04:56:00

%S 1,3,37,997,44121,2882071,260415373,31088448777,4737782756017,

%T 897380763253291,206773800208348341,56951114596754707693,

%U 18476855531112777659017,6973886287904020598308287,3029760395576715276955711261,1501087423496953812426438796561

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

%H 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; also arXiv preprint arXiv:1605.00172, 2016.

%H Bernd Sturmfels, <a href="http://www.ams.org/publications/journals/notices/201606/rnoti-p604.pdf">Tensors and Their Eigenvalues</a>, Notices AMS, 63 (No. 6, 2016), 606-606. (Th. 9 gives g.f.)

%p ans:=[];

%p for d from 1 to 10 do

%p for h from 1 to d do zh[h]:=add(z[i],i=1..d)-z[h]; od;

%p t1:= expand(simplify( mul( (zh[i]^3-z[i]^3) / (zh[i]-z[i]), i=1..d)));

%p a:=t1; for i from 1 to d do a:=coeff(a,z[i],2); od;

%p ans:=[op(ans),a];

%p od:

%p ans;

%t 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]] ];

%t Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 1, 12}] (* _Jean-François Alcover_, Nov 26 2018 *)

%o (PARI)

%o P(n, t='t) = {

%o   my(z=vector(n, k, eval(Str("z", k))),

%o      s1=sum(k=1, #z, z[k]), s2=sum(k=1, #z, z[k]^2), s12=(s1^2 - s2)/2,

%o      f=vector(n, k, s2 + t*(s12 - z[k]*(s1 - z[k])) + z[k]*(s1 - z[k])), g=1);

%o   for (i=1, n, g *= f[i]; for(j=1, n, g=substpol(g, z[j]^3, 0)));

%o   for (k=1, n, g=polcoef(g, 2, z[k]));

%o   g;

%o };

%o vector(10, n, P(n,2)) \\ _Gheorghe Coserea_, Nov 27 2018

%Y Row n=3 of A284308.

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

%K nonn

%O 1,2

%A _N. J. A. Sloane_, Jun 21 2016

%E a(11)-a(15) from _Gheorghe Coserea_, Jun 29 2016

%E a(16) from _Alois P. Heinz_, Mar 24 2017