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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: A244127 A342312 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 February 8 23:03 EST 2023. Contains 360153 sequences. (Running on oeis4.)