A028338 → x ↦ −x+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:
  0:         -1,           1;
  1:          6,          -5,          1;
  2:        -40,          38,        -11,           1;
  3:        336,        -356,        128,         -19,         1;
  4:      -3456,        3984,      -1660,         320,       -29,         1;
  5:      42240,      -52064,      24184,       -5540,       670,       -41,       1;
  6:    -599040,      779904,    -394240,      102424,    -14980,      1246,     -55,       1;
  7:    9676800,   -13190400,    7139712,    -2044784,    343504,    -35000,    2128,     -71,    1;
  8: -175472640,   248776704, -142545664,    44162432,  -8259216,    976080,  -73416,    3408,  -89,    1;
  9: 3530096640, -5179060224, 3114809856, -1030100480, 210065120, -27853392, 2448768, -141720, 5190, -109, 1;

The row polynomials satisfy (2*n-1-x)*P(n,x) = -(4*n^2+2*n-(4*n+1)*x+x^2)*P(n-1,x) for n >= 1, with P(0,x) = -1 + x.

P[0, x_] := -1 + x;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n^2 + 2*n - (4*n + 1)*x + x^2)*P[n - 1, x])/(2*n - 1 - x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = -1 + x; for(n = 1, N, my(i = n + 1); v[i] = (-(4*n^2 + 2*n - (4*n + 1)*x + x^2)*v[i - 1])/(2*n - 1 - x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

Proved properties

• Real-rooted. Every row P_n, n >= 0, 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-769dd7c64d51b4a5

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-769dd7c64d51b4a5

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

  • premise (sequence): Signed double Pochhammer triangle: expansion of x(x-2)(x-4)..(x-2n+2).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
    • preserver_image affine, shift 0 e-8b470702f19ec444

      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): Signed double Pochhammer triangle: expansion of x(x-2)(x-4)..(x-2n+2).
    uses: Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1
    • ↵ preserver_image affine, shift 0 e-8b470702f19ec444 (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.