A028338 → x ↦ (−2/9)x+1 (shift +2) → 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:                 -4212,                   864,                 -53,                   1;
  1:                153090,                -35559,                2781,                 -89,                 1;
  2:              -6928416,               1761264,             -161190,                6795,              -134,                1;
  3:             375551640,            -102364722,            10491039,             -528930,             14040,             -188,              1;
  4:          -23723526240,            6840708552,          -764776404,            43875594,          -1414665,            25893,           -251,             1;
  5:         1711537626960,         -517192003908,         62000505216,         -3928641507,         145859049,         -3280662,          43974,          -323,          1;
  6:      -138851503257600,        43666593782400,      -5546211809616,        380620169964,      -15756084036,        411827409,       -6844824,         70146,       -404,         1;
  7:     12512256087300480,     -4073562175547664,     543390640077996,     -39837784582620,     1799993324205,     -52850311542,     1028259603,     -13160880,     106515,      -494,      1;
  8:  -1239978636966182400,    416192628042434880,  -57919716432985968,    4490792652763296,  -218180254026240,    7035910374240,  -154710013689,    2331820953,  -23709510,    155430,   -593,    1;
  9: 134031568381456838400, -46225842241704994080, 6676462999455338088, -543287324331671436, 28070603157862386, -978537963455415, 23753908190529, -406665758763, 4893410313, -40500405, 219483, -701, 1;

The row polynomials satisfy (9*n+17-x)*P(n,x) = -(81*n^2+396*n+468-(18*n+44)*x+x^2)*P(n-1,x) for n >= 1, with P(0,x) = -4212 + 864*x - 53*x^2 + x^3.

P[0, x_] := -4212 + 864*x - 53*x^2 + x^3;
P[n_, x_] := P[n, x] = Expand[Together[(-(81*n^2 + 396*n + 468 - (18*n + 44)*x + x^2)*P[n - 1, x])/(9*n + 17 - x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten

P(N) = {my(v = vector(N + 1)); v[1] = -4212 + 864*x - 53*x^2 + x^3; for(n = 1, N, my(i = n + 1); v[i] = (-(81*n^2 + 396*n + 468 - (18*n + 44)*x + x^2)*v[i - 1])/(9*n + 17 - 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-07f8adddfb9eb09c

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-07f8adddfb9eb09c

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

  • premise (sequence): Triangle read by rows: T(n, m) = S1(n, m)*9^(n-m), where S1 are the signed Stirling numbers of first kind A008275 (n >=…
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
    • preserver_image affine, shift 2 e-2bc76bcaa23d62fb

      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 >= 2, 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 read by rows: T(n, m) = S1(n, m)*9^(n-m), where S1 are the signed Stirling numbers of first kind A008275 (n >=…
    uses: Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1
    • ↵ preserver_image affine, shift 2 e-2bc76bcaa23d62fb (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.