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A321711 Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section. 2
1, 1, 0, 3, 0, 0, 11, 9, 0, 1, 53, 120, 60, 40, 9, 309, 1410, 1800, 1590, 885, 216, 2119, 16560, 39960, 55120, 52065, 29016, 7570, 16687, 202755, 801780, 1696555, 2433165, 2300403, 1326850, 357435, 148329, 2624496, 15606360, 48387024, 99650670, 141429456, 135382464, 79738800, 22040361, 1468457, 36080100, 304274880, 1323453180, 3760709526, 7493549868, 10570597800, 10199809980, 6103007505, 1721632024 (list; table; graph; refs; listen; history; text; internal format)
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

Cf. A000255, A000681, A007107, A274308, A284989.

Sequence in context: A177016 A244127 A123474 * A277788 A208848 A277945

Adjacent sequences:  A321708 A321709 A321710 * A321712 A321713 A321714

KEYWORD

nonn,tabl

AUTHOR

Gheorghe Coserea, Nov 27 2018

STATUS

approved

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Last modified November 12 07:16 EST 2019. Contains 329052 sequences. (Running on oeis4.)