A028338 → x ↦ −2x+9 → 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:         -4,           1;
  1:         25,         -10,          1;
  2:       -180,          96,        -17,          1;
  3:       1470,        -959,        233,        -25,         1;
  4:     -13440,       10208,      -3074,        459,       -34,         1;
  5:     136080,     -116586,      41199,      -7690,       800,       -44,       1;
  6:   -1512000,     1429800,    -573100,     126274,    -16525,      1285,     -55,      1;
  7:   18295200,   -18797460,    8351524,   -2096563,    325409,    -31990,    1946,    -67,    1;
  8: -239500800,   264219840, -127984896,   35741292,  -6344240,    742609,  -57344,   2818,  -80,   1;
  9: 3372969600, -3958933680, 2064977532, -630593996, 124903421, -16774086, 1547427, -96864, 3939, -94, 1;

The row polynomials satisfy (n+3-x)*P(n,x) = -(n^2+8*n+16-(2*n+8)*x+x^2)*P(n-1,x) for n >= 1, with P(0,x) = -4 + x.

P[0, x_] := -4 + x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^2 + 8*n + 16 - (2*n + 8)*x + x^2)*P[n - 1, x])/(n + 3 - x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = -4 + x; for(n = 1, N, my(i = n + 1); v[i] = (-(n^2 + 8*n + 16 - (2*n + 8)*x + x^2)*v[i - 1])/(n + 3 - 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-6d01d329dd41619f

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-6d01d329dd41619f

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

  • premise (sequence): Generalized Stirling number triangle of first kind read by rows: T(n, k) = [x^k] Product_{m=1..n} (x - m - r), with r =…
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
    • preserver_image affine, shift 0 e-6b900047034f4def

      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): Generalized Stirling number triangle of first kind read by rows: T(n, k) = [x^k] Product_{m=1..n} (x - m - r), with r =…
    uses: Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1
    • ↵ preserver_image affine, shift 0 e-6b900047034f4def (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.