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
Keywords
Links
- 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.)
Programs
-
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 *) -
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
Extensions
a(11)-a(15) from Gheorghe Coserea, Jun 29 2016
a(16) from Alois P. Heinz, Mar 24 2017