A028338 → x ↦ −2x+17 → 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:           -8,             1;
  1:           81,           -18,           1;
  2:         -900,           280,         -29,           1;
  3:        10890,         -4279,         629,         -41,          1;
  4:      -142560,         66816,      -12486,        1163,        -54,          1;
  5:      2007720,      -1082562,      242279,      -28810,       1920,        -68,       1;
  6:    -30270240,      18317544,    -4729892,      675634,     -57665,       2941,     -83,       1;
  7:    486486000,    -324504900,    94251240,   -15570051,    1600249,    -104790,    4270,     -99,    1;
  8:  -8302694400,    6022540800, -1931761296,   359647756,  -42833924,    3384689, -177464,    5954, -116,    1;
  9: 149967417600, -117052405200, 40893480684, -8421696604, 1132318277, -103866126, 6583395, -284736, 8043, -134, 1;

The row polynomials satisfy (n+7-x)*P(n,x) = -(n^2+16*n+64-(2*n+16)*x+x^2)*P(n-1,x) for n >= 1, with P(0,x) = -8 + x.

P[0, x_] := -8 + x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^2 + 16*n + 64 - (2*n + 16)*x + x^2)*P[n - 1, x])/(n + 7 - x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = -8 + x; for(n = 1, N, my(i = n + 1); v[i] = (-(n^2 + 16*n + 64 - (2*n + 16)*x + x^2)*v[i - 1])/(n + 7 - 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-319662bfd9739006

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-319662bfd9739006

  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-86838a696acab184

      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-86838a696acab184 (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.