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A321711 Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section.

%I A321711 #28 Dec 24 2018 21:31:02
%S A321711 1,1,0,3,0,0,11,9,0,1,53,120,60,40,9,309,1410,1800,1590,885,216,2119,
%T A321711 16560,39960,55120,52065,29016,7570,16687,202755,801780,1696555,
%U A321711 2433165,2300403,1326850,357435,148329,2624496,15606360,48387024,99650670,141429456,135382464,79738800,22040361,1468457,36080100,304274880,1323453180,3760709526,7493549868,10570597800,10199809980,6103007505,1721632024
%N A321711 Triangle T(n,k) read by rows: coefficients of polynomials P_n(t) defined in Formula section.
%H A321711 Gheorghe Coserea, <a href="/A321711/b321711.txt">Rows n = 0..13, flattened</a>
%H A321711 Shmuel Friedland, Giorgio Ottaviani, <a href="https://arxiv.org/abs/1210.8316">The number of singular vector tuples and uniqueness of best rank one approximation of tensors</a>, arXiv:1210.8316 [math.AG], 2013.
%F A321711 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.
%F A321711 A000255(n) = T(n,0).
%F A321711 A007107(n) = T(n,n).
%F A321711 A000681(n) = Sum_{k=0..n} T(n,k).
%F A321711 A274308(n) = Sum_{k=0..n} T(n,k)*2^k.
%e A321711 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.
%e A321711 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 + ...
%e A321711 Triangle starts:
%e A321711 n\k [0]    [1]     [2]     [3]      [4]      [5]      [6]      [7]
%e A321711 [0] 1;
%e A321711 [1] 1;     0;
%e A321711 [2] 3;     0;      0;
%e A321711 [3] 11,    9,      0,      1;
%e A321711 [4] 53,    120,    60,     40,      9;
%e A321711 [5] 309,   1410,   1800,   1590,    885,     216;
%e A321711 [6] 2119,  16560,  39960,  55120,   52065,   29016,   7570;
%e A321711 [7] 16687, 202755, 801780, 1696555, 2433165, 2300403, 1326850, 357435;
%e A321711 [8] ...
%o A321711 (PARI)
%o A321711 P(n, t='t) = {
%o A321711   my(z=vector(n, k, eval(Str("z", k))),
%o A321711      s1=sum(k=1, #z, z[k]), s2=sum(k=1, #z, z[k]^2), s12=(s1^2 - s2)/2,
%o A321711      f=vector(n, k, s2 + t*(s12 - z[k]*(s1 - z[k])) + z[k]*(s1 - z[k])), g=1);
%o A321711   for (i=1, n, g *= f[i]; for(j=1, n, g=substpol(g, z[j]^3, 0)));
%o A321711   for (k=1, n, g=polcoef(g, 2, z[k]));
%o A321711   g;
%o A321711 };
%o A321711 seq(N) = concat([[1], [1, 0], [3, 0, 0]], apply(n->Vecrev(P(n,'t)), [3..N]));
%o A321711 concat(seq(9))
%Y A321711 Cf. A000255, A000681, A007107, A274308, A284989.
%K A321711 nonn,tabl
%O A321711 0,4
%A A321711 _Gheorghe Coserea_, Nov 27 2018