A028338 → x ↦ −2x−3 → Pₙ + Pₙ₊₁

OEIS matches ?

• A167891: row sums, 14 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS
• A167876: row sums, 15 terms agree OEIS nonn (row-sum match: finite agreement only) · view at the OEIS

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: 2,     1;
  1: 1,     2,      1;
  2: 0,     0,      1,    1;
  3: 0,     1,     -1,   -1,     1;
  4: 0,    -4,      4,    3,    -4,      1;
  5: 0,    18,    -21,  -10,    20,     -8,     1;
  6: 0,   -96,    128,   34,  -115,     61,   -13,     1;
  7: 0,   600,   -890,  -91,   749,   -490,   140,   -19,   1;
  8: 0, -4320,   6984, -204, -5474,   4249, -1484,   274, -26,   1;
  9: 0, 35280, -61236, 8476, 44477, -40026, 16275, -3696, 483, -34, 1;

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

P[0, x_] := 2 + x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^2 - 4*n + 4 - (2*n - 4)*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] = 2 + x; for(n = 1, N, my(i = n + 1); v[i] = (-(n^2 - 4*n + 4 - (2*n - 4)*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-167ee17f7f3ebd58

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-167ee17f7f3ebd58

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

  • premise (sequence): Expansion of e.g.f. (1 + y)^(1 + x).
    uses: Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)
    • preserver_image affine, shift 0 e-6cd09736036c3bec

      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): Expansion of e.g.f. (1 + y)^(1 + x).
    uses: Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1
    • ↵ preserver_image affine, shift 0 e-6cd09736036c3bec (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.