OFFSET
0,4
LINKS
Gheorghe Coserea, Rows n = 0..13, flattened
Shmuel Friedland, Giorgio Ottaviani, The number of singular vector tuples and uniqueness of best rank one approximation of tensors, arXiv:1210.8316 [math.AG], 2013.
FORMULA
Let z1..zn be n variables and s1 = Sum_{k=1..n} zk, s2 = Sum_{k=1..n} zk^2, s12 = (s1^2 - s2)/2, fk = s2 + t*(s12 - zk*(s1 - zk)) + zk*(s1 - zk) for k=1..n; we define P_n(t) = [(z1..zn)^2] Product_{k=1..n} fk.
A000255(n) = T(n,0).
A007107(n) = T(n,n).
A000681(n) = Sum_{k=0..n} T(n,k).
A274308(n) = Sum_{k=0..n} T(n,k)*2^k.
EXAMPLE
For n=3 we have s1 = z1 + z2 + z3, s2 = z1^2 + z2^2 + z3^2, s12 = z1*z2 + z1*z3 + z2*z3, f1 = z1^2 + z2^2 + z3^2 + t*z2*z3 + z1*(z2 + z3), f2 = z1^2 + z2^2 + z3^2 + t*z1*z3 + z2*(z1 + z3), f3 = z1^2 + z2^2 + z3^2 + t*z1*z2 + z3*(z1 + z2), [(z1*z2*z3)^2] f1*f2*f3 = 11 + 9*t + t^3, therefore P_3(t) = 11 + 9*t + t^3.
A(x;t) = 1 + x + 3*x^2 + (11 + 9*t + t^3)*x^3 + (53 + 120*t + 60*t^2 + 40*t^3 + 9*t^4)*x^4 + ...
Triangle starts:
n\k [0] [1] [2] [3] [4] [5] [6] [7]
[0] 1;
[1] 1; 0;
[2] 3; 0; 0;
[3] 11, 9, 0, 1;
[4] 53, 120, 60, 40, 9;
[5] 309, 1410, 1800, 1590, 885, 216;
[6] 2119, 16560, 39960, 55120, 52065, 29016, 7570;
[7] 16687, 202755, 801780, 1696555, 2433165, 2300403, 1326850, 357435;
[8] ...
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;
};
seq(N) = concat([[1], [1, 0], [3, 0, 0]], apply(n->Vecrev(P(n, 't)), [3..N]));
concat(seq(9))
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Gheorghe Coserea, Nov 27 2018
STATUS
approved