A028338 → x ↦ −x (shift -1) → Pₙ + Pₙ₊₁

OEIS matches ?

No OEIS matches.

OEIS submission text ?

Fields to paste into an OEIS submission: the example %e, the formula %F and programs %t (Mathematica) and %o (PARI/GP) that list the same rows. For an OEIS link the row numbers, the offset and the orientation are those of the entry, as matched; the matched terms were compared with the published ones, which identifies the node with the entry on a finite prefix, not for every n. The PARI/GP program is not machine-checked here.

Triangle begins:
   1:         0,           1;
   2:         2,          -3,          1;
   3:       -12,          19,         -8,          1;
   4:        90,        -153,         77,        -15,         1;
   5:      -840,        1513,       -864,        214,       -24,         1;
   6:      9450,      -17835,      11189,      -3250,       480,       -35,       1;
   7:   -124740,      244683,    -165192,      53899,     -9540,       937,     -48,       1;
   8:   1891890,    -3834033,    2746841,    -980637,    198009,    -23667,    1659,     -63,    1;
   9: -32432400,    67598865,  -50884992,   19532428,  -4366880,    602294,  -51968,    2732,  -80,   1;
  10: 620269650, -1325007315, 1040262057, -424077156, 102919256, -15859746, 1593102, -104004, 4254, -99, 1;

The row polynomials satisfy (2*n-4-x)*P(n,x) = -(4*n^2-10*n+6-(4*n-5)*x+x^2)*P(n-1,x) for n >= 2, with P(1,x) = x.

P[1, x_] := x;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n^2 - 10*n + 6 - (4*n - 5)*x + x^2)*P[n - 1, x])/(2*n - 4 - x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten

P(N) = {my(v = vector(N)); v[1] = x; for(n = 2, N, my(i = n); v[i] = (-(4*n^2 - 10*n + 6 - (4*n - 5)*x + x^2)*v[i - 1])/(2*n - 4 - x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

Proved properties

• Real-rooted. Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, +inf)
  Every polynomial of the sequence has only real roots, inside the stated interval.
  by the cited steps: e-557516b36262f3b9

Proof chains

1 incoming proof step. Each step is expanded back to the roots of the proof DAG, the theorem checks of a node's own rule. JSON

• obreschkoff_combination e-557516b36262f3b9

  Hermite–Kakeya–Obreschkoff theorem [Obr63; Fis06, Thm 1.43] ?

  • premise (sequence): Triangle of coefficients in expansion of (x-1)*(x-3)*(x-5)*...*(x-(2*n-1)).
    uses: Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, +inf)
    • preserver_image affine, shift -1 e-d7c9e146299b18aa

      Affine, reversal and Möbius substitutions (elementary) polytool ?

      • premise (sequence): Triangle of coefficients in expansion of (x+1)*(x+3)*...*(x + 2n - 1) in rising powers of x.
        uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]
        • theorem_check liu_wang e-59b849691f316c6f root: theorem check

          Liu–Wang theorem [LW07, Thm 2.1, Cor 2.4] ?

        • parent_provers root_domain e-ef36674c01230a86 root: theorem check

          Parent-row provers of polytool polytool ?

  • premise (sequence): Triangle of coefficients in expansion of (x-1)*(x-3)*(x-5)*...*(x-(2*n-1)).
    uses: Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2
    • ↵ preserver_image affine, shift -1 e-d7c9e146299b18aa (expanded above)

Related sequences ?

No candidate equalities or related sequences.

References

1. [Fis06] Steve Fisk. Polynomials, roots, and interlacing. arXiv:math/0612833. arXiv
2. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
3. [Obr63] Nikola Obreschkoff. Verteilung und berechnung der nullstellen reeller polynome. Hochschulbücher für mathematik, VEB Deutscher Verlag der Wissenschaften, 1963.