cp's OEIS Frontend

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

%I A274308 #33 Dec 01 2018 04:56:00
%S A274308 1,3,37,997,44121,2882071,260415373,31088448777,4737782756017,
%T A274308 897380763253291,206773800208348341,56951114596754707693,
%U A274308 18476855531112777659017,6973886287904020598308287,3029760395576715276955711261,1501087423496953812426438796561
%N A274308 Number of n-tuples of singular vectors of a 3 X 3 X 3 X ... X 3 n-dimensional tensor.
%H A274308 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 A274308 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 A274308 ans:=[];
%p A274308 for d from 1 to 10 do
%p A274308 for h from 1 to d do zh[h]:=add(z[i],i=1..d)-z[h]; od;
%p A274308 t1:= expand(simplify( mul( (zh[i]^3-z[i]^3) / (zh[i]-z[i]), i=1..d)));
%p A274308 a:=t1; for i from 1 to d do a:=coeff(a,z[i],2); od;
%p A274308 ans:=[op(ans),a];
%p A274308 od:
%p A274308 ans;
%t A274308 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 A274308 Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 1, 12}] (* _Jean-François Alcover_, Nov 26 2018 *)
%o A274308 (PARI)
%o A274308 P(n, t='t) = {
%o A274308   my(z=vector(n, k, eval(Str("z", k))),
%o A274308      s1=sum(k=1, #z, z[k]), s2=sum(k=1, #z, z[k]^2), s12=(s1^2 - s2)/2,
%o A274308      f=vector(n, k, s2 + t*(s12 - z[k]*(s1 - z[k])) + z[k]*(s1 - z[k])), g=1);
%o A274308   for (i=1, n, g *= f[i]; for(j=1, n, g=substpol(g, z[j]^3, 0)));
%o A274308   for (k=1, n, g=polcoef(g, 2, z[k]));
%o A274308   g;
%o A274308 };
%o A274308 vector(10, n, P(n,2)) \\ _Gheorghe Coserea_, Nov 27 2018
%Y A274308 Row n=3 of A284308.
%Y A274308 Cf. A271905, A272551, A283829, A283830, A321711.
%K A274308 nonn
%O A274308 1,2
%A A274308 _N. J. A. Sloane_, Jun 21 2016
%E A274308 a(11)-a(15) from _Gheorghe Coserea_, Jun 29 2016
%E A274308 a(16) from _Alois P. Heinz_, Mar 24 2017